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ON  THE 


STUDY  AND  DIFFICULTIES  OF 
MATHEMATICS 


AUGUSTUS  DE  MORGAN 


SECOND  REPRINT  EDITION 


CHICAGO 
THE  OPEN  COURT  PUBLISHING  COMPANY 

LONDON 

KEGAN  PAUL,  TRENCH,  TRUBNER  &  Co.,  LTD, 
1902 


IN  THE  SAME  SERIES 


GEOMETRIC  EXERCISES  IN  PAPER-FOLDING.  By  T.  SUNDARA 
Row.  Edited  and  revised  by  W.  W.  Bentan  and  D.  E.  Smith. 
With  many  half-tone  engravings  from  photographs  of  actual  ex- 
ercises, and  a  package  of  papers  for  folding.  Pages,  x,  148.  Cloth, 
Si.oo  net  (45.  6d.  net). 

LECTURES  ON  ELEMENTARY  MATHEMATICS.  By  JOSKPH 
Louis  LAGRANGE.  From  the  French  by  Thomas  J.  McCormack. 
With  portrait  and  biography.  Pages,  172.  Second  edition.  Cloth, 
fi.oo  net  (45.  6d.  net). 

ESSAYS  ON  THE  THEORY  OF  NUMBERS:  (i)  ON  CONTINUITY 
AND  IRRATIONAL  NUMBERS  ;  (2)  ON  THE  NATURE  AND  MEANING  OF 
NUMBERS.  By  R.  DEDEKIND.  From  the  German  by  W.  W.  Beman. 
Pages,  115.  Cloth,  75  cents  net  (33.  net). 

ON  THE  STUDY  AND  DIFFICULTIES  OF  MATHEMATICS.  By 
AUGUSTUS  DE  MORGAN.  Second  reprint  edition,  with  portrait 
and  bibliographies.  Pp.,  288.  Cloth,  $1.25  net  (45.  6d.  net). 

ELEMENTARY  ILLUSTRATIONS  OF  THE  DIFFERENTIAL 
AND  INTEGRAL  CALCULUS.  By  AUGUSTUS  DE  MORGAN.  Re- 
print edition.  With  a  bibliography  of  text-books  of  the  Calculus. 
Pp.,  144.  Price,  Ji.oo  net  (43.  6d.  net). 

MATHEMATICAL  ESSAYS  AND  RECREATIONS.  By  PROF.  HER- 
MANN SCHUBERT,  of  Hamburg,  Germany.  From  the  German  by 
T.  J.  McCormack.  Essays  on  Number,  The  Magic  Square,  The 
Fourth  Dimension,  The  Squaring  of  the  Circle.  Pp.,  149.  Price, 
Cloth,  75c.  net  (33.  net). 

A  BRIEF  HISTORY  OF  ELEMENTARY  MATHEMATICS.  By 
ETR.  KARL  FINK,  of  Tubingen.  From  the  German  by  W.  W.  Beman 
and  D.  E.  Smith.  Pp.,  333.  Cloth,  81.50  net  (53.  6d.  net). 

THE  FOUNDATIONS  OF  GEOMETRY.  By  DAVID  HILBERT, 
Ph.  D.,  Professor  of  Mathematics  in  the  University  of  Gottingen. 
Authorised  translation  by  E.  J.  Townsend,  Ph.  D.  (In  preparation.) 


THE  OPEN  COURT  PUBLISHING  COMPANY 

324    DEARBORN    ST.,    CHICAGO. 

LONDON:  Kegan  Paul,  Trench,  Triibner  &  Co.,  Ltd. 


/  I 


EDITOR'S  NOTE. 

A  TO  apology  is  needed  for  the  publication  of  the  present  new 
-*-^  edition  of  The  Study  and  Difficulties  of  Mathematics, — a 
characteristic  production  of  one  of  the  most  eminent  and  lumi- 
nous of  English  mathematical  writers  of  the  present  century.  De 
Morgan,  though  taking  higher  rank  as  an  original  inquirer  than 
either  Huxley  or  Tyndall,  was  the  peer  and  lineal  precursor  of 
these  great  expositors  of  science,  and  he  applied  to  his  lifelong  task 
an  historical  equipment  and  a  psychological  insight  which  have 
not  yet  borne  their  full  educational  fruit.  And  nowhere  have  these 
distinguished  qualities  been  displayed  to  greater  advantage  than  in 
the  present  work,  which  was  conceived  and  written  with  the  full 
natural  freedom,  and  with  all  the  fire,  of  youthful  genius.  For  the 
contents  and  purpose  of  the  book  the  reader  may  be  referred  to 
the  Author's  Preface.  The  work  still  contains  points  (notable 
among  them  is  its  insistence  on  the  study  of  logic),  which  are  in- 
sufficiently emphasised,  or  slurred,  by  elementary  treatises ;  while 
the  freshness  and  naturalness  of  its  point  of  view  contrasts  strongly 
with  the  mechanical  character  of  the  common  text-books.  Ele- 
mentary instructors  and  students  cannot  fail  to  profit  by  the  gen- 
eral loftiness  of  its  tone  and  the  sound  tenor  of  its  instructions. 

The  original  treatise,  which  was  published  by  the  Society  for 
the  Diffusion  of  Useful  Knowledge  and  bears  the  date  of  1831,  is 
now  practically  inaccessible,  and  is  marred  by  numerous  errata 
and  typographical  solecisms,  from  which,  it  is  hoped,  the  present 
edition  is  free.  References  to  the  remaining  mathematical  text- 
books of  the  Society  for  the  Diffusion  of  Useful  Knowledge  now 


IV  ON  THE  STUDY  OF  MATHEMATICS. 

out  of  print  have  either  been  omitted  or  supplemented  by  the  men- 
tion of  more  modern  works.  The  few  notes  which  have  been 
added  are  mainly  bibliographical  in  character,  and  refer,  for  in- 
stance, to  modern  treatises  on  logic,  algebra,  the  philosophy  of 
mathematics,  and  pangeometry.  For  the  portrait  and  autograph 
signature  of  De  Morgan,  which  graces  the  page  opposite  the  title, 
The  Open  Court  Publishing  Company  is  indebted  to  the  courtesy 
of  Principal  David  Eugene  Smith,  of  the  State  Normal  School  at 
Brockport,  N.  Y. 

THOMAS  J.  McCoRMACK. 
LA  SALLE,  111.,  Nov.  i,  1898. 


AUTHOR'S  PREFACE. 

'N  compiling  the  following  pages,  my  object  has  been  to  notice 
particularly  several  points  in  the  principles  of  algebra  and 
geometry,  which  have  not  obtained  their  due  importance  in  our 
jmentary  works  on  these  sciences.    There  are  two  classes  of  men 

fho  might  be  benefited  by  a  work  of  this  kind,  viz. ,  teachers  of 
the  elements,  who  have  hitherto  confined  their  pupils  to  the  work- 
ing of  rules,  without  demonstration,  and  students,  who,  having 
acquired  some  knowledge  under  this  system,  find  their  further 
progress  checked  by  the  insufficiency  of  their  previous  methods 
and  attainments.  To  such  it  must  be  an  irksome  task  to  recom- 
mence their  studies  entirely  ;  I  have  therefore  placed  before  them, 
by  itself,  the  part  which  has  been  omitted  in  their  mathematical 
education,  presuming  throughout  in  my  reader  such  a  knowledge 
of  the  rules  of  algebra,  and  the  theorems  of  Euclid,  as  is  usually 
obtained  in  schools. 

It  is  needless  to  say  that  those  who  have  the  advantage  of 
University  education  will  not  find  more  in  this  treatise  than  a  little 
thought  would  enable  them  to  collect  from  the  best  works  now  in 
use  [1831],  both  at  Cambridge  and  Oxford.  Nor  do  I  pretend  to 
settle  the  many  disputed  points  on  which  I  have  necessarily  been 
obliged  to  treat.  The  perusal  of  the  opinions  of  an  individual, 
offered  simply  as  such,  may  excite  many  to  become  inquirers,  who 
would  otherwise  have  been  workers  of  rules  and  followers  of  dog- 
mas. They  may  not  ultimately  coincide  in  the  views  promulgated 
by  the  work  which  first  drew  their  attention,  but  the  benefit  which 
they  will  derive  from  it  is  not  the  less  on  that  account.  I  am  not, 


vi  ON  THE  STUDY  OF  MATHEMATICS. 

however,  responsible  for  the  contents  of  this  treatise,  further  than 
for  the  manner  in  which  they  are  presented,  as  most  of  the  opin- 
ions here  maintained  have  been  found  in  the  writings  of  eminent 
mathematicians. 

It  has  been  my  endeavor  to  avoid  entering  into  the  purely 
metaphysical  part  of  the  difficulties  of  algebra.  The  student  is,  in 
my  opinion,  little  the  better  for  such  discussions,  though  he  may 
derive  such  conviction  of  the  truth  of  results  by  deduction  from 
particular  cases,  as  no  &  priori  reasoning  can  give  to  a  beginner. 
In  treating,  therefore,  on  the  negative  sign,  on  impossible  quanti- 
ties, and  on  fractions  of  the  form  §,  etc.,  I  have  followed  the 
method  adopted  by  several  of  the  most  esteemed  continental  writ- 
ers, of  referring  the  explanation  to  some  particular  problem,  and 
showing  how  to  gain  the  same  from  any  other.  Those  who  admit 
such  expressions  as  — a,  */ — a,  g,  etc.,  have  never  produced  any 
clearer  method ;  while  those  who  call  them  absurdities,  and  would 
reject  them  altogether,  must,  I  think,  be  forced  to  admit  the  fact 
that  in  algebra  the  different  species  of  contradictions  in  problems 
are  attended  with  distinct  absurdities,  resulting  from  them  as 
necessarily  as  different  numerical  results  from  different  numerical 
data.  This  being  granted,  the  whole  of  the  ninth  chapter  of  this 
work  may  be  considered  as  an  inquiry  into  the  nature  of  the  differ- 
ent misconceptions,  which  give  rise  to  the  various  expressions 
above  alluded  to.  To  this  view  of  the  question  I  have  leaned, 
finding  no  other  so  satisfactory  to  my  own  mind. 

The  number  of  mathematical  students,  increased  as  it  has 
been  of  late  years,  would  be  much  augmented  if  those  who  hold 
the  highest  rank  in  science  would  condescend  to  give  more  effective 
assistance  in  clearing  the  elements  of  the  difficulties  which  they 
present.  If  any  one  claiming  that  title  should  think  my  attempt 
obscure  or  erroneous,  he  must  share  the  blame  with  me,  since  it  is 
through  his  neglect  that  I  have  been  enabled  to  avail  myself  of  an 
opportunity  to  perform  a  task  which  I  would  gladly  have  seen  con- 
fided to  more  skilful  hands.  AUGUSTUS  DE  MORGAN. 


CONTENTS. 


CHAPTER  PAGE 

Editor's  Note iii 

Author's  Preface v 

I.  Introductory  Remarks  on  the  Nature  and  Objects  of 

Mathematics i 

II.  On  Arithmetical  Notation n 

III.  Elementary  Rules  of  Arithmetic .20 

IV.  Arithmetical  Fractions .'     .     .     .  30 

V.  Decimal  Fractions 42 

VI.  Algebraical  Notation  and  Principles 55 

VII.  Elementary  Rules  of  Algebra 67 

VIII.  Equations  of  the  First  Degree 90 

IX.  On  the  Negative  Sign,  etc 103 

X.  Equations  of  the  Second  Degree 129 

XI.  On  Roots  in  General,  and  Logarithms 158 

XII.   On  the  Study  of  Algebra 175 

XIII.  On  the  Definitions  of  Geometry 191 

XIV.  On  Geometrical  Reasoning 203 

XV.  On  Axioms 231 

XVI.  On  Proportion 240 

XVII.  Application  of  Algebra  to  the  Measurement  of  Lines, 

Angles,  Proportion  of  Figures,  and  Surfaces.     .     .  266 


CHAPTER  I. 

INTRODUCTORY  REMARKS  ON  THE  NATURE  AND 
OBJECTS  OF  MATHEMATICS. 

THE  OBJECT  of  this  Treatise  is— (1)  To  point 
out  to  the  student  of  Mathematics,  who  has  not 
the  advantage  of  a  tutor,  the  course  of  study  which  it 
is  most  advisable  that  he  should  follow,  the  extent  to 
which  he  should  pursue  one  part  of  the  science  before 
he  commences  another,  and  to  direct  him  as  to  the 
sort  of  applications  which  he  should  make.  (2)  To 
treat  fully  of  the  various  points  which  involve  difficul- 
ties and  which  are  apt  to  be  misunderstood  by  begin- 
ners, and  to  describe  at  length  the  nature  without 
going  into  the  routine  of  the  operations. 

No  person  commences  the  study  of  mathematics 
without  soon  discovering  that  it  is  of  a  very  different 
nature  from  those  to  which  he  has  been  accustomed. 
The  pursuits  to  which  the  mind  is  usually  directed  be- 
fore entering  on  the  sciences  of  algebra  and  geometry, 
are  such  as  languages  and  history,  etc.  Of  these, 
neither  appears  to  have  any  affinity  with  mathemat- 


2  ON  THE  STUDY  OF  MATHEMATICS. 

ics ;  yet,  in  order  to  see  the  difference  which  exists  be- 
tween these  studies, — for  instance,  history  and  geom- 
etry,— it  will  be  useful  to  ask  how  we  come  by  knowl- 
edge in  each.  Suppose,  for  example,  we  feel  certain 
of  a  fact  related  in  history,  such  as  the  murder  of 
Caesar,  whence  did  we  derive  the  certainty?  how  came 
we  to  feel  sure  of  the  general  truth  of  the  circum- 
stances of  the  narrative?  The  ready  answer  to  this 
question  will  be,  that  we  have  not  absolute  certainty 
upon  this  point ;  but  that  we  have  the  relation  of  his- 
torians, men  of  credit,  who  lived  and  published  their 
accounts  in  the  very  time  of  which  they  write  ;  that 
succeeding  ages  have  received  those  accounts  as  true, 
and  that  succeeding  historians  have  backed  them  with 
a  mass  of  circumstantial  evidence  which  makes  it  the 
most  improbable  thing  in  the  world  that  the  account, 
or  any  material  part  of  it,  should  be  false.  This  is 
perfectly  correct,  nor  can  there  be  the  slightest  ob- 
jection to  believing  the  whole  narration  upon  such 
grounds ;  nay,  our  minds  are  so  constituted,  that, 
upon  our  knowledge  of  these  arguments,  we  cannot 
help  believing,  in  spite  of  ourselves.  But  this  brings 
us  to  the  point  to  which  we  wish  to  come  ;  we  believe 
that  Caesar  was  assassinated  by  Brutus  and  his  friends, 
not  because  there  is  any  absurdity  in  supposing  the 
contrary,  since  every  one  must  allow  that  there  is  just 
a  possibility  that  the  event  never  happened  :  not  be- 
cause we  can  show  that  it  must  necessarily  have  been 
that,  at  a  particular  day,  at  a  particular  place,  a  sue- 


NATURE  AND  OBJECTS  OF  MATHEMATICS.  3 

cessful  adventurer  must  have  been  murdered  in  the 
manner  described,  but  because  our  evidence  of  the 
fact  is  such,  that,  if  we  apply  the  notions  of  evidence 
which  every-day  experience  justifies  us  in  entertain- 
ing, we  feel  that  the  improbability  of  the  contrary 
compels  us  to  take  refuge  in  the  belief  of  the  fact ; 
and,  if  we  allow  that  there  is  still  a  possibility  of  its 
falsehood,  it  is  because  this  supposition  does  not  in- 
volve absolute  absurdity,  but  only  extreme  improb- 
ability. 

In  mathematics  the  case  is  wholly  different.  It  is 
true  that  the  facts  asserted  in  these  sciences  are  of  a 
nature  totally  distinct  from  those  of  history  ;  so  much 
so,  that  a  comparison  of  the  evidence  of  the  two  may 
almost  excite  a  smile.  But  if  it  be  remembered  that 
acute  reasoners,  in  every  branch  of  learning,  have 
acknowledged  the  use,  we  might  almost  say  the  neces- 
sity, of  a  mathematical  education,  it  must  be  admitted 
that  the  points  of  connexion  between  these  pursuits 
and  others  are  worth  attending  to.  They  are  the  more 
so,  because  there  is  a  mistake  into  which  several  have 
fallen,  and  have  deceived  others,  and  perhaps  them- 
selves, by  clothing  some  false  reasoning  in  what  they 
called  a  mathematical  dress,  imagining  that,  by  the 
application  of  mathematical  symbols  to  their  subject, 
they  secured  mathematical  argument.  This  could  not 
have  happened  if  they  had  possessed  a  knowledge  of 
the  bounds  within  which  the  empire  of  mathematics 
is  contained.  That  empire  is  sufficiently  wide,  and 


4  ON  THE  STUDY  OF  MATHEMATICS. 

might  have  been  better  known,  had  the  time  which 
has  been  wasted  in  aggressions  upon  the  domains  of 
others,  been  spent  in  exploring  the  immense  tracts 
which  are  yet  untrodden. 

We  have  said  that  the  nature  of  mathematical  dem- 
onstration is  totally  different  from  all  other,  and  the 
difference  consists  in  this — that,  instead  of  showing 
the  contrary  of  the  proposition  asserted  to  be  only  im- 
probable, it  proves  it  at  once  to  be  absurd  and  impos- 
sible. This  is  done  by  showing  that  the  contrary  of 
the  proposition  which  is  asserted  is  in  direct  contra- 
diction to  some  extremely  evident  fact,  of  the  truth  of 
which  our  eyes  and  hands  convince  us.  In  geometry, 
of  the  principles  alluded  to,  those  which  are  most 
commonly  used  are — 

I.  If  a  magnitude  be  divided  into  parts,  the  whole 
is  greater  than  either  of  those  parts. 

II.  Two  straight  lines  cannot  inclose  a  space. 

III.  Through  one  point  only  one  straight  line  can 
be  drawn,  which  never  meets  another  straight  line,  or 
which  is  parallel  to  it. 

It  is  on  such  principles  as  these  that  the  whole  of 
geometry  is  founded,  and  the  demonstration  of  every 
proposition  consists  in  proving  the  contrary  of  it  to  be 
inconsistent  with  one  of  these.  Thus,  in  Euclid,  Book 
I.,  Prop.  4,  it  is  shown  that  two  triangles  which  have 
two  sides  and  the  included  angle  respectively  equal 
are  equal  in  all  respects,  by  proving  that,  if  they  are 
not  equal,  two  straight  lines  will  inclose  a  space,  which 


NATURE  AND  OBJECTS  OF  MATHEMATICS.  5 

is  impossible.  In  other  treatises  on  geometry,  the 
same  thing  is  proved  in  the  same  way,  only  the  self- 
evident  truth  asserted  sometimes  differs  in  form  from 
that  of  Euclid,  but  may  be  deduced  from  it,  thus — 

Two  straight  lines  which  pass  through  the  same 
two  points  must  either  inclose  a  space,  or  coincide 
and  be  one  and  the  same  line,  but  they  cannot  inclose 
a  space,  therefore  they  must  coincide.  Either  of  these 
propositions  being  granted,  the  other  follows  imme- 
diately ;  it  is,  therefore,  immaterial  which  of  them  we 
use.  We  shall  return  to  this  subject  in  treating 
specially  of  the  first  principles  of  geometry. 

Such  being  the  nature  of  mathematical  demonstra- 
tion, what  we  have  before  asserted  is  evident,  that 
our  assurance  of  a  geometrical  truth  is  of  a  nature 
wholly  distinct  from  that  which  we  can  by  any  means 
obtain  of  a  fact  in  history  or  an  asserted  truth  of  meta- 
physics. In  reality,  our  senses  are  our  first  mathe- 
matical instructors;  they  furnish  us  with  notions 
which  we  cannot  trace  any  further  or  represent  in  any 
other  way  than  by  using  single  words,  which  every 
one  understands.  Of  this  nature  are  the  ideas  to 
which  we  attach  the  terms  number,  one,  two,  three, 
etc.,  point,  straight  line,  surface;  all  of  which,  let 
them  be  ever  so  much  explained,  can  never  be  made 
any  clearer  than  they  are  already  to  a  child  of  ten 
years  old. 

But,  besides  this,  our  senses  also  furnish  us  with 
the  means  of  reasoning  on  the  things  which  we  call 


6  ON    THE    STUDY    OF    MATHEMATICS. 

by  these  names,  in  the  shape  of  incontrovertible  prop- 
ositions, such  as  have  been  already  cited,  on  which, 
if  any  remark  is  made  by  the  beginner  in  mathemat- 
ics, it  will  probably  be,  that  from  such  absurd  truisms 
as  "the  whole  is  greater  than  its  part,"  no  useful  re- 
sult can  possibly  be  derived,  and  that  we  might  as 
well  expect  to  make  use  of  ' '  two  and  two  make  four. " 
This  observation,  which  is  common  enough  in  the 
mouths  of  those  who  are  commencing  geometry,  is 
the  result  of  a  little  pride,  which  does  not  quite  like 
the  humble  operation  of  beginning  at  the  beginning, 
and  is  rather  shocked  at  being  supposed  to  want  such 
elementary  information.  But  it  is  wanted,  neverthe- 
less ;  the  lowest  steps  of  a  ladder  are  as  useful  as  the 
highest.  Now,  the  most  common  reflection  on  the 
nature  of  the  propositions  referred  to  will  convince  us 
of  their  truth.  But  they  must  be  presented  to  the  un- 
derstanding, and  reflected  on  by  it,  since,  simple  as 
they  are,  it  must  be  a  mind  of  a  very  superior  cast 
which  could  by  itself  embody  these  axioms,  and  pro- 
ceed from  them  only  one  step  in  the  road  pointed  out 
in  any  treatise  on  geometry. 

But,  although  there  is  no  study  which  presents  so 
simple  a  beginning  as  that  of  geometry,  there  is  none 
in  which  difficulties  grow  more  rapidly  as  we  proceed, 
and  what  may  appear  at  first  rather  paradoxical,  the 
more  acute  the  student  the  more  serious  will  the  im- 
pediments in  the  way  of  his  progress  appear.  This 
necessarily  follows  in  a  science  which  consists  of  rea- 


NATURE  AND  OBJECTS  OF  MATHEMATICS.  7 

soning  from  the  very  commencement,  for  it  is  evident 
that  every  student  will  feel  a  claim  to  have  his  objec- 
tions answered,  not  by  authority,  but  by  argument, 
and  that  the  intelligent  student  will  perceive  more 
readily  than  another  the  force  of  an  objection  and  the 
obscurity  arising  from  an  unexplained  difficulty,  as 
the  greater  is  the  ordinary  light  the  more  will  occa- 
sional darkness  be  felt.  To  remove  some  of  these 
difficulties  is  the  principal  object  of  this  Treatise. 

We  shall  now  make  a  few  remarks  on  the  advan- 
tages to  be  derived  from  the  study  of  mathematics, 
considered  both  as  a  discipline  for  the  mind  and  a  key 
to  the  attainment  of  other  sciences.  It  is  admitted  by 
all  that  a  finished  or  even  a  competent  reasoner  is  not 
the  work  of  nature  alone  ;  the  experience  of  every  day 
makes  it  evident  that  education  develops  faculties 
which  would  otherwise  never  have  manifested  their 
existence.  It  is,  therefore,  as  necessary  to  learn  to 
reason  before  we  can  expect  to  be  able  to  reason,  as  it 
is  to  learn  to  swim  or  fence,  in  order  to  attain  either 
of  those  arts.  Now,  something  must  be  reasoned 
upon,  it  matters  not  much  what  it  is,  provided  that  it 
can  be  reasoned  upon  with  certainty.  The  properties 
of  mind  or  matter,  or  the  study  of  languages,  mathe- 
matics, or  natural  history,  may  be  chosen  for  this  pur- 
pose. Now,  of  all  these,  it  is  desirable  to  choose  the 
one  which  admits  of  the  reasoning  being  verified,  that 
is,  in  which  we  can  find  out  by  other  means,  such  as 
measurement  and  ocular  demonstration  of  all  sorts, 


8  ON   THE    STUDY   OF   MATHEMATICS, 

whether  the  results  are  true  or  not,  When  the  guid- 
ing property  of  the  loadstone  was  first  ascertained, 
and  it  was  necessary  to  learn  how  to  use  this  new  dis- 
covery, and  to  find  out  how  far  it  might  be  relied  on, 
it  would  have  been  thought  advisable  to  make  many 
passages  between  ports  that  were  well  known  before 
attempting  a  voyage  of  discovery.  So  it  is  with  our 
reasoning  faculties  :  it  is  desirable  that  their  powers 
should  be  exerted  upon  objects  of  such  a  nature,  that 
we  can  tell  by  other  means  whether  the  results  which 
we  obtain  are  true  or  false,  and  this  before  it  is  safe 
to  trust  entirely  to  reason.  Now  the  mathematics  are 
peculiarly  well  adapted  for  this  purpose,  on  the  fol- 
lowing grounds : 

1.  Every  term  is  distinctly  explained,  and  has  but 
one  meaning,  and  it  is  rarely  that  two  words  are  em- 
ployed to  mean  the  same  thing. 

2.  The  first  principles  are  self-evident,  and,  though 
derived  from  observation,  do  not  require  more  of  it 
than  has  been  made  by  children  in  general. 

3.  The    demonstration    is    strictly   logical,    taking 
nothing  for  granted  except  the  self-evident  first  prin- 
ciples, resting  nothing  upon  probability,  and  entirely 
independent  of  authority  and  opinion. 

4.  When  the  conclusion  is  attained  by  reasoning, 
its  truth  or  falsehood  can  be  ascertained,  in  geometry 
by  actual  measurement,  in  algebra  by  common  arith- 
metical calculation.     This  gives   confidence,    and  is. 


NATURE  AND  OBJECTS  OF  MATHEMATICS.  9 

absolutely  necessary,  if,  as  was  said  before,  reason  is 
not  to  be  the  instructor,  but  the  pupil. 

5.  There  are  no  words  whose  meanings  are  so 
much  alike  that  the  ideas  which  they  stand  for  may 
be  confounded.  Between  the  meanings  of  terms  there 
is  no  distinction,  except  a  total  distinction,  and  all 
adjectives  and  adverbs  expressing  difference  of  de- 
grees are  avoided.  Thus  it  may  be  necessary  to  say, 
"A  is  greater  than  B;"  but  it  is  entirely  unimportant 
whether  A  is  very  little  or  very  much  greater  than  B. 
Any  proposition  which  includes  the  foregoing  asser- 
tion will  prove  its  conclusion  generally,  that  is,  for  all 
cases  in  which  A  is  greater  than  B,  whether  the  dif- 
ference be  great  or  little.  Locke  mentions  the  dis- 
tinctness of  mathematical  terms,  and  says  in  illustra- 
tion :  "The  idea  of  two  is  as  distinct  from  the  idea  of 
"three  as  the  magnitude  of  the  whole  earth  is  from 
"that  of  a  mite.  This  is  not  so  in  other  simple  modes, 
"in  which  it  is  not  so  easy,  nor  perhaps  possible  for  us 
"to  distinguish  between  two  approaching  ideas,  which 
"yet  are  really  different ;  for  who  will  undertake  to 
"find  a  difference  between  the  white  of  this  paper, 
"and  that  of  the  next  degree  to  it  ?" 

These  are  the  principal  grounds  on  which,  in  our 
opinion,  the  utility  of  mathematical  studies  may  be 
shown  to  rest,  as  a  discipline  for  the  reasoning  pow- 
ers. But  the  habits  of  mind  which  these  studies  have 
a  tendency  to  form  are  valuable  in  the  highest  degree. 
The  most  important  of  all  is  the  power  of  concentrat- 


10  ON  THE  STUDY  OF  MATHEMATICS. 

ing  the  ideas  which  a  successful  study  of  them  in- 
creases where  it  did  exist,  and  creates  where  it  did 
not.  A  difficult  position,  or  a  new  method  of  passing 
from  one  proposition  to  another,  arrests  all  the  atten- 
tion and  forces  the  united  faculties  to  use  their  utmost 
exertions.  The  habit  of  mind  thus  formed  soon  ex- 
tends itself  to  other  pursuits,  and  is  beneficially  felt 
in  all  the  business  of  life. 

As  a  key  to  the  attainment  of  other  sciences,  the 
use  of  the  mathematics  is  too  well  known  to  make  it 
necessary  that  we  should  dwell  on  this  topic.  In  fact, 
there  is  not  in  this  country  any  disposition  to  under- 
value them  as  regards  the  utility  of  their  applications. 
But  though  they  are  now  generally  considered  as  a 
part,  and  a  necessary  one,  of  a  liberal  education,  the 
views  which  are  still  taken  of  them  as  a  part  of  edu- 
cation by  a  large  proportion  of  the  community  are 
still  very  confined. 

The  elements  of  mathematics  usually  taught  are 
contained  in  the  sciences  of  arithmetic,  algebra,  geom- 
etry, and  trigonometry.  We  have  used  these  four  di- 
visions because  they  are  generally  adopted,  though, 
in  fact,  algebra  and  geometry  are  the  only  two  of  them 
which  are  really  distinct.  Of  these  we  shall  commence 
with  arithmetic,  and  take  the  others  in  succession  in 
the  order  in  which  we  have  arranged  them. 


CHAPTER  II. 

ON  ARITHMETICAL  NOTATION. 

r  I  ^HE  first  ideas  of  arithmetic,  as  well  as  those  of 
-^-  other  sciences,  are  derived  from  early  observa- 
tion. How  they  come  into  the  mind  it  is  unnecessary 
to  inquire ;  nor  is  it  possible  to  define  what  we  mean 
by  number  and  quantity.  They  are  terms  so  simple, 
that  is,  the  ideas  which  they  stand  for  are  so  com- 
pletely the  first  ideas  of  our  mind,  that  it  is  impossible 
to  find  others  more  simple,  by  which  we  may  explain 
them.  This  is  what  is  meant  by  defining  a  term ;  and 
here  we  may  say  a  few  words  on  definitions  in  general, 
which  will  apply  equally  to  all  sciences. 

Definition  is  the  explaining  a  term  by  means  of 
others,  which  are  more  easily  understood,  and  thereby 
fixing  its  meaning,  so  that  it  may  be  distinctly  seen 
what  it  does  imply,  as  well  as  what  it  does  not.  Great 
care  must  be  taken  that  the  definition  itself  is  not  a 
tacit  assumption  of  some  fact  or  other  which  ought  to 
be  proved.  Thus,  when  it  is  said  that  a  square  is  "a 
four-sided  figure,  all  whose  sides  are  equal,  and  all 


12  ON  THE  STUDY  OF  MATHEMATICS. 

whose  angles  are  right  angles,"  though  no  more  is 
said  than  is  true  of  a  square,  yet  more  is  said  than  is 
necessary  to  define  it,  because  it  can  be  proved  that 
if  a  four-sided  figure  have  all  its  sides  equal,  and  one 
only  of  its  angles  a  right  angle,  all  the  other  angles 
must  be  right  angles  also.  Therefore,  in  making  the 
above  definition,  we  do,  in  fact,  affirm  that  which 
ought  to  be  proved.  Again,  the  above  definition, 
though  redundant  in  one  point,  is,  strictly  speaking, 
defective  in  another,  for  it  omits  to  state  whether  the 
sides  of  the  figure  are  straight  lines  or  curves.  It 
should  be,  "a  square  is  a  four-sided  rectilinear  figure, 
all  of  whose  sides  are  equal,  and  one  of  whose  angles 
is  a  right  angle." 

As  the  mathematical  sciences  owe  much,  if  not  all, 
of  the  superiority  of  their  demonstrations  to  the  pre- 
cision with  which  the  terms  are  defined,  it  is  most  es- 
sential that  the  beginner  should  see  clearly  in  what  a 
good  definition  consists.  We  have  seen  that  there 
are  terms  which  cannot  be  defined,  such  as  number 
and  quantity.  An  attempt  at  a  definition  would  only 
throw  a  difficulty  in  the  student's  way,  which  is  already 
done  in  geometry  by  the  attempts  at  an  explanation 
of  the  terms  point,  straight  line,  and  others,  which 
are  to  be  found  in  treatises  on  that  subject.  A  point  is 
defined  to  be  that  "  which  has  no  parts,  and  which 
has  no  magnitude";  a  straight  line  is  that  which 
"lies  evenly  between  its  extreme  points."  Now,  let 
any  one  ask  himself  whether  he  could  have  guessed 


ON  ARITHMETICAL  NOTATION.  1 3 

what  was  meant,  if,  before  he  began  geometry,  any 
one  had  talked  to  him  of  "that  which  has  no  parts 
and  which  has  no  magnitude,"  and  "the  line  which 
lies  evenly  between  its  extreme  points,"  unless  he  had 
at  the  same  time  mentioned  the  words  "point"  and 
"straight  line/'  which  would  have  removed  the  diffi- 
culty? In  this  case  the  explanation  is  a  great  deal 
harder  than  the  term  to  be  explained,  which  must 
always  happen  whenever  we  are  guilty  of  the  absurd- 
ity of  attempting  to  make  the  simplest  ideas  yet  more 
simple. 

A  knowledge  of  our  method  of  reckoning,  and  of 
writing  down  numbers,  is  taught  so  early,  that  the 
method  by  which  we  began  is  hardly  recollected. 
Few,  therefore,  reflect  upon  the  very  commencement 
of  arithmetic,  or  upon  the  simplicity  and  elegance 
with  which  calculations  are  conducted.  We  find  the 
method  of  reckoning  by  ten  in  our  hands,  we  hardly 
know  how,  and  we  conclude,  so  natural  and  obvious 
does  it  seem,  that  it  came  with  our  language,  and  is 
a  part  of  it ;  and  that  we  are  not  much  indebted  to 
instruction  for  so  simple  a  gift.  It  has  been  well  ob- 
served, that  if  the  whole  earth  spoke  the  same  lan- 
guage, we  should  think  that  the  name  of  any  object 
was  not  a  mere  sign  chosen  to  represent  it,  but  was  a 
sound  which  had  some  real  connexion  with  the  thing ; 
and  that  we  should  laugh  at,  and  perhaps  persecute, 
any  one  who  asserted  that  any  other  sound  would  do 
as  well  if  we  chose  to  think  so.  We  cannot  fall  into 


14  ON  THE  STUDY  OF  MATHEMATICS. 

this  error,  because,  as  it  is,  we  happen  to  know  that 
what  we  call  by  the  sound  "horse,"  the  Romans  dis- 
tinguished as  well  by  that  of  "equus"  but  we  commit 
a  similar  mistake  with  regard  to  our  system  of  nume- 
ration, because  at  present  it  happens  to  be  received 
by  all  civilised  nations,  and  we  do  not  reflect  on  what 
was  done  formerly  by  almost  all  the  world,  and  is  done 
still  by  savages.  The  following  considerations  will, 
perhaps,  put  this  matter  on  a  right  footing,  and  show 
that  in  our  ideas  of  arithmetic  we  have  not  altogether 
rid  ourselves  of  the  tendency  to  attach  ideas  of  mysti- 
cism to  numbers  which  has  prevailed  so  extensively 
in  all  times. 

We  know  that  we  have  nine  signs  to  stand  for  the 
first  nine  numbers,  and  one  for  nothing,  or  zero.  Also, 
that  to  represent  ten  we  do  not  use  a  new  sign,  but 
combine  two  of  the  others,  and  denote  it  by  10,  eleven 
by  11,  and  so  on.  But  why  was  the  number  ten  chosen 
as  the  limit  of  our  separate  symbols — why  not  nine, 
eight,  or  eleven?  If  we  recollect  how  apt  we  are  to 
count  on  the  fingers,  we  shall  be  at  no  loss  to  see  the 
reason.  We  can  imagine  our  system  of  numeration 
formed  thus  : — A  man  proceeds  to  count  a  number, 
and  to  help  the  memory  he  puts  a  finger  on  the  table 
for  each  one  which  he  counts.  He  can  thus  go  as  far 
as  ten,  after  which  he  must  begin  again,  and  by  reck- 
oning the  fingers  a  second  time  he  will  have  counted 
twenty,  and  so  on.  But  this  is  not  enough  ;  he  must 
also  reckon  the  number  of  times  which  he  has  done 


ON    ARITHMETICAL    NOTATION.  15 

this,  and  as  by  counting  on  the  fingers  he  has  divided 
the  things  which  he  is  counting  into  lots  of  ten  each, 
he  may  consider  each  lot  as  a  unit  of  its  kind,  just  as 
we  say  a  number  of  sheep  is  one  flock,  twenty  shillings 
are  one  pound.  Call  each  lot  a  ten.  In  this  way  he 
can  count  a  ten  of  tens,  which  he  may  call  a  hundred, 
a  ten  of  hundreds,  or  a  thousand,  and  so  on.  The 
process  of  reckoning  would  then  be  as  follows: — Sup- 
pose, to  choose  an  example,  a  number  of  faggots  is  to 
be  counted.  They  are  first  tied  up  in  bundles  of  ten 
each,  until  there  are  not  so  many  as  ten  left.  Suppose 
there  are  seven  over.  We  then  count  the  bundles  of 
ten  as  we  counted  the  single  faggots,  and  tie  them  up 
also  by  tens,  forming  new  bundles  of  one  hundred 
each  with  some  bundles  of  ten  remaining.  Let  these 
last  be  six  in  number.  We  then  tie  up  the  bundles 
of  hundreds  by  tens,  making  bundles  of  thousands, 
and  find  that  there  are  five  bundles  of  hundreds  re- 
maining. Suppose  that  on  attempting  to  tie  up  the 
thousands  by  tens,  we  find  there  are  not  so  many  as 
ten,  but  only  four.  The  number  of  faggots  is  then  4 
thousands,  5  hundreds,  6  tens,  and  7. 

The  next  question  is,  how  shall  we  represent  this 
number  in  a  short  and  convenient  manner?  It  is  plain 
that  the  way  to  do  this  is  &  matter  of  choice.  Suppose 
then  that  we  distinguish  the  tens  by  marking  their 
number  with  one  accent,  the  hundreds  with  two  ac- 
cents, and  the  thousands  with  three.  We  may  then 
represent  this  number  in  any  of  the  following  ways  : — 


l6  OK  THE  STUDY  OF  MATHEMATICS. 

76'5"4"',  6'75"4'",  6'4'"5"7,  4"'5"6'7,  the  whole  num- 
ber of  ways  being  24.  But  this  is  more  than  we  want  ; 
one  certain  method  of  representing  a  number  is  suffi- 
cient. The  most  natural  way  is  to  place  them  in  order 
of  magnitude,  either  putting  the  largest  collection  first 
or  the  smallest ;  thus  4"'5"6'7,  or  76'5"4"p.  Of  these 
we  choose  the  first. 

In  writing  down  numbers  in  this  way  it  will  soon 
be  apparent  that  the  accents  are  unnecessary.  Since 
the  singly  accented  figure  will  always  be  the  second 
from  the  right,  and  so  on,  the  place  of  each  number 
will  point  out  what  accents  to  write  over  it,  and  we 
may  therefore  consider  each  figure  as  deriving  a  value 
from  the  place  in  which  it  stands.  But  here  this  diffi- 
culty occurs.  How  are  we  to  represent  the  numbers 
3'"3',  and  4"'2'7  without  accents?  If  we  write  them 
thus,  33  and  427,  they  will  be  mistaken  for  3'3  and 
4"2'7.  This  difficulty  will  be  obviated  by  placing  cy- 
phers so  as  to  bring  each  number  into  the  place  al- 
lotted to  the  sort  of  collection  which  it  represents  ; 
thus,  since  the  trebly  accented  letters,  or  thousands, 
are  in  the  fourth  place  from  the  right,  and  the  singly 
accented  letters  in  the  second,  the  first  number  may 
be  written  3030,  and  the  second  4027.  The  cypher, 
which  plays  so  important  a  part  in  arithmetic  that  it 
was  anciently  called  the  art  of  cypher,  or  cyphering, 
does  not  stand  for  any  number  in  itself,  but  is  merely 
employed,  like  blank  types  in  printing,  to  keep  other 
signs  in  those  places  which  they  must  occupy  in  order 


ON  ARITHMETICAL  NOTATION.  17 

to  be  read  rightly.  We  may  now  ask  what  would 
have  been  the  case  if,  instead  of  ten  fingers,  men  had 
had  more  or  less.  For  example,  by  what  signs  would 
4567  have  been  represented,  if  man  had  nine  fingers 
instead  of  ten?  We  may  presume  that  the  method 
would  have  been  the  same,  with  the  number  nine  rep- 
resented by  10  instead  of  ten,  and  the  omission  of  the 
symbol  9.  Suppose  this  number  of  faggots  is  to  be 
counted  by  nines.  Tie  them  up  in  bundles  of  nine, 
and  we  shall  find  4  faggots  remaining.  Tie  these 
bundles  again  in  bundles  of  nine,  each  of  which  will, 
therefore,  contain  eighty-one,  and  there  will  be  3  bun- 
dles remaining.  These  tied  up  in  the  same,  way  into 
bundles  of  nine,  each  of  which  contains  seven  hundred 
and  twenty-nine,  will  leave  2  odd  bundles,  and,  as 
there  will  be  only  six  of  them,  the  process  cannot  be 
carried  any  further.  If,  then,  we  represent,  by  1',  a 
bundle  of  nine,  or  a  nine,  by  1"  a  nine  of  nines,  and 
so  on,  the  number  which  we  write  4567,  must  be  writ- 
ten 6'"  2"  3'  4.  In  order  to  avoid  confusion,  we  will 
suffer  the  accents  to  remain  over  all  numbers  which 
are  not  reckoned  in  tens,  while  those  which  are  so 
reckoned  shall  be  written  in  the  common  way.  The 
following  is  a  comparison  of  the  way  in  which  num- 
bers in  the  common  system  are  written,  and  in  the 
one  which  we  have  just  explained  : 

COUNTING  BY 

Tens...l  2  3  4  5  6  7  8  9  10  11  12  13 
Nines..  1  2  3  4  5  6  7  8  I'O  11  1'2  1'3  T4 


1 8  ON  THE  STUDY  OF  MATHEMATICS. 

COUNTING  BY 

Tens 14  15  16  17  18  19  20  30  40  50 

Nines 1'5  1'6  VI  1'8  2'0  2'1  2'2  3'3  4'4  5'5 

Tens 60     70     80       90       100 

Nines 6'6    7'7    8'8    1"1'0    1"2'1 

We  will  now  write,  in  the  common  way,  in  the 
tens'  system,  the  process  which  we  went  through  in 
order  to  find  how  to  represent  the  number  4567  in 
that  of  the  nines,  thus  : 

9)4567 

9)  507  — rem.  4. 
9)  56  —  rem.  3. 
9)  6  — rem.  2. 

0  — rem.  6.      Representation  required,  6'"  2"  3' 4. 

The  processes  of  arithmetic  are  the  same  in  prin- 
ciple whatever  system  of  numeration  is  used.  To 
show  this,  we  subjoin  a  question  in  each  of  the  first 
four  rules,  worked  both  in  the  common  system,  and 
in  that  of  the  nines.  There  is  the  difference,  that,  in 
the  first,  the  tens  must  be  carried,  and  in  the  second 
the  nines. 

ADDITION. 

636  7"  7' 6 

987  1"'3"1'6 

403  4"  8' 7 

2026  2"'7"0'1 

SUBTRACTION. 

1384  r"8"0'7 

797  1"'0"7'5 

587  T  2'  2 


ON   ARITHMETICAL  NOTATION. 
MULTIPLICATION. 

297  3"  6'   0 

136  1"  6'   1 


1782  360 

891  2400 

297  360 


40392  6""!'"  3"  6'   0 

DIVISION. 

633)  79125  (125  7"  7'  3)  lv  S^O1"^1?1  6*  (1"4'  8 
633  773 

1582  4217 

1266  3423 


6846 
6846 

0 

The  student  should  accustom  himself  to  work 
questions  in  different  systems  of  numeration,  which 
will  give  him  a  clearer  insight  into  the  nature  of  arith- 
metical processes  than  he  could  obtain  by  any  other 
method.  When  he  uses  a  system  in  which  numbers 
are  counted  by  a  number  greater  than  ten,  he  will 
want  some  new  symbols  for  figures.  For  example,  in 
the  duodecimal  system,  where  twelve  is  the  number 
of  figures  supposed,  twelve  will  be  represented  by  TO ; 
there  must,  therefore,  be  a  distinct  sign  for  ten  and 
eleven  :  a  nine  and  six  reversed,  thus  9  and  d,  might 
be  used-  for  these. 

*To  avoid  too  great  a  number  of  accents,  Roman  numerals  are  put  in- 
stead of  them;  also,  to  avoid  confusion,  the  accents  are  omitted  after  the 
first  line. 


CHAPTER  III. 

ELEMENTARY  RULES  OF  ARITHMETIC. 

AS  SOON  as  the  beginner  has  mastered  the  notion 
•*"*•  of  arithmetic,  he  may  be  made  acquainted  with 
the  meaning  of  the  algebraical  signs  -f,  — ,  X>  =,  and 
also  with  that  for  division,  or  the  common  way  of  rep- 
resenting a  fraction.  There  is  no  difficulty  in  these 
signs  or  in  their  use.  Five  minutes'  consideration  will 
make  the  symbol  5  -j-  3  present  as  clear  an  idea  as  the 
words  "5  added  to  3."  The  reason  why  they  usually 
cause  so  much  embarrassment  is,  that  they  are  gener- 
ally deferred  until  the  student  commences  algebra, 
when  he  is  often  introduced  at  the  same  time  to  the 
representation  of  numbers  by  letters,  the  distinction 
of  known  and  unknown  quantities,  the  signs  of  which 
we  have  been  speaking,  and  the  use  of  figures  as 
the  exponents  of  letters.  Either  of  these  four  things 
is  quite  sufficient  at  a  time,  and  there  is  no  time  more 
favorable  for  beginning  to  make  use  of  the  signs  of 
operation  than  when  the  habit  of  performing  the  ope- 
rations commences.  The  beginner  should  exercise 


ELEMENTARY  RULES  OF  ARITHMETIC.  21 

himself  in  putting  the  simplest  truths  of  arithmetic  in 
this  new  shape,  and  should  write  such  sentences  as 
the  following  frequently  : 


6  —  4  =  2, 

4  —  6  =  4  +  2  +  1, 
2  X  2  +  12  X  12  =  1  4  x  10  +  2  X  2  X  2. 

These  will  accustom  him  to  the  meaning  of  the  signs, 
just  as  he  was  accustomed  to  the  formation  of  letters 
by  writing  copies.  As  he  proceeds  through  the  rules 
of  arithmetic,  he  should  take  care  never  to  omit  con- 
necting each  operation  with  its  sign,  and  should  avoid 
confounding  operations  together  and  considering  them 
as  the  same,  because  they  produce  the  same  result. 
Thus  4  X  ~  does  not  denote  the  same  operation  as 
7X4,  though  the  result  of  both  is  28.  The  first  is 
four  multiplied  by  seven,  four  taken  seven  times  ;  the 
second  is  seven  multiplied  by  four,  seven  taken  four 
times;  and  that  4x7  =  7x4  is  a  proposition  to  be 
proved,  not  to  be  taken  for  granted.  Again,  \  X  4 
and  ^  are  marks  of  distinct  operations,  though  their 
result  is  the  same,  as  we  shall  show  in  treating  of 
fractions. 

The  examples  which  a  beginner  should  choose  for 
practice  should  be  simple  and  should  not  contain  very 
large  numbers.  The  powers  of  the  mind  cannot  be 
directed  to  two  things  at  once  :  if  the  complexity  of 
the  numbers  used  requires  all  the  student's  attention, 
he  cannot  observe  the  principle  of  the  rule  which  he 


22  ON  THE  STUDY  OF  MATHEMATICS. 

is  following.  Now,  at  the  commencement  of  his  ca- 
reer, a  principle  is  not  received  and  understood  by 
the  student  as  quickly  as  it  is  explained  by  the  in- 
structor. He  does  not,  and  cannot,  generalise  at  all ; 
he  must  be  taught  to  do  so ;  and  he  cannot  learn  that 
a  particular  fact  holds  good  for  all  numbers  unless  by 
having  it  shown  that  it  holds  good  for  some  numbers, 
and  that  for  those  some  numbers  he  may  substitute 
others,  and  use  the  same  demonstration.  Until  he 
can  do  this  himself  he  does  not  understand  the  prin- 
ciple, and  he  can  never  do  this  except  by  seeing  the 
rule  explained  and  trying  it  himself  on  small  numbers. 
He  may,  indeed,  and  will,  believe  it  on  the  word  of 
his  instructor,  but  this  disposition  is  to  be  checked. 
He  must  be  told,  that  whatever  is  not  gained  by  his 
own  thought  is  not  gained  to  any  purpose ;  that  the 
mathematics  are  put  in  his  way  purposely  because 
they  are  the  only  sciences  in  which  he  must  not  trust 
the  authority  of  any  one.  The  superintendence  of 
these  efforts  is  the  real  business  of  an  instructor  in 
arithmetic.  The  merely  showing  the  student  a  rule 
by  which  he  is  to  work,  and  comparing  his  answer 
with  a  key  to  the  book,  printed  for  the  preceptor's 
private  use,  to  save  the  trouble  which  he  ought  to 
bestow  upon  his  pupil,  is  not  teaching  arithmetic  any 
more  than  presenting  him  with  a  grammar  and  dic- 
tionary is  teaching  him  Latin.  When  the  principle 
of  each  rule  has  been  well  established  by  showing  its 
application  to  some  simple  examples  (and  the  number 


ELEMENTARY  RULES  OF  ARITHMETIC.  23 

of  these  requisite  will  vary  with  the  intellect  of  the 
student),  he  may  then  proceed  to  more  complicated 
cases,  in  order  to  acquire  facility  in  computation.  The 
four  first  rules  may  be  studied  in  this  way,  and  these 
will  throw  the  greatest  light  on  those  which  succeed. 

The  student  must  observe  that  all  operations  in 
arithmetic  may  be  resolved  into  addition  and  subtrac- 
tion ;  that  these  additions  and  subtractions  might  be 
made  with  counters ;  so  that  the  whole  of  the  rules 
consist  of  processes  intended  to  shorten  and  simplify 
that  which  would  otherwise  be  long  and  complex.  For 
example,  multiplication  is  continued  addition  of  the 
same  number  to  itself — twelve  times  seven  is  twelve 
sevens  added  together.  Division  is  a  continued  sub- 
traction of  one  number  from  another;  the  division  of 
129  by  3  is  a  continued  subtraction  of  3  from  129,  in 
order  to  see  how  many  threes  it  contains.  All  other 
operations  are  composed  of  these  four,  and  are,  there- 
fore, the  result  of  additions  and  subtractions  only. 

The  following  principles,  which  occur  so  continu- 
ally in  mathematical  operations  that  we  are,  at  length, 
hardly  sensible  of  their  presence,  are  the  foundation 
of  the  arithmetical  rules : 

I.  We  do  not  alter  the  sum  of  two  numbers  by 
taking  away  any  part  of  the  first,  if  we  annex  that 
part  to  the  second.  This  may  be  expressed  by  signs, 
in  a  particular  instance,  thus : 

(20  —  6)  +  (32  +  6)  =  20  +  32. 


24  ON  THE  STUDY  OF  MATHEMATICS. 

II.  We  do  not  alter  the  difference  of  two  numbers 
by  increasing  or  diminishing  one  of  them,  provided 
we  increase  or  diminish  the  other  as  much.    This  may 
be  expressed  thus,  in  one  instance : 

(45  +  7)— (22 +  7)==  46— 22. 

(45  —  8)  —  (22  —  8)  =  45  —  22. 

III.  If  we  wish  to  multiply  one  number  by  another, 
for  example  156  by  29,  we  may  break  up  156  into  any 
number  of  parts,  multiply  each  of  these  parts  by  29 
and  add  the  results.     For  example,  156  is  made  up  of 
100,  50,  and  6.     Then 

156  X  29  =  100  x  29  -|-  50  X  29  +  6  X  29. 

IV.  The  same  thing  may  be  done  with  the  multi- 
plier instead  of  the  multiplicand.     Thus,  29  is  made 
up  of  18,  6,  and  5.      Then 

156  X  29  =  156  X  18  +  156  X  6  +  156  X  5. 

V.  If  any  two  or  more  numbers  be  multiplied  to- 
gether, it  is  indifferent  in  what  order  they  are  multi- 
plied, the  result  is  the  same.     Thus, 

10X6X4X3  =  3X10X4X6  =  6X10X4X3,  etc. 

VI.  In  dividing  one  number  by  another,  for  ex- 
ample 156  by  12,  we  may  break  up  the  dividend,  and 
divide  each  of  its  parts  by  the  divisor,  and  then  add 
the  results.     We  may  part  156  into  72,  60,  and  24; 
this  is  expressed  thus  : 

156_72        60        24 


ELEMENTARY  RULES  OF  ARITHMETIC.  25 

The  same  thing  cannot  be  done  with  the  divisor.      It 
is  not  true  that 

156  _     156        156         156 
~~~~     ~~       ~~~       ~~' 


The  student  should  discover  the  reason  for  himself. 

A  prime  number  is  one  which  is  not  divisible  by 
any  other  number  except  1.  When  the  process  of  di- 
vision can  be  performed,  it  can  be  ascertained  whether 
a  given  number  is  divisible  by  any  other  number,  that 
is,  whether  it  is  prime  or  not.  This  can  be  done  by 
dividing  it  by  all  the  numbers  which  are  less  than  its 
half,  since  it  is  evident  that  it  cannot  be  divided  into 
a  number  of  parts,  each  of  which  is  greater  than  its 
half.  This  process  would  be  laborious  when  the  given 
number  is  large  ;  still  it  may  be  done,  and  by  this 
means  the  number  itself  may  be  reduced  to  its  prime 
factors*  as  it  is  called,  that  is,  it  may  either  be  shown 
to  be  a  prime  number  itself  or  made  up  by  multiply- 
ing several  prime  numbers  together.  Thus,  306  is 
34  X  9,  or  2  X  17  X  9,  or  2x17x3x3,  and  has  for 
its  prime  factors  2,  17,  and  3,  the  latter  of  which  is 
repeated  twice  in  its  formation.  When  this  has  been 
done  with  two  numbers,  we  can  then  see  whether 
they  have  any  factors  in  common,  and,  if  that  be  the 
case,  we  can  then  find  what  is  called  their  greatest 
common  measure  or  divisor;  that  is,  the  number  made 


*The  factors  of  a  number  are  those  numbers  by  the  multiplication  of 
which  it  is  made. 


26  ON  THE  STUDY  OF  MATHEMATICS. 

by  multiplying  all  their  common  factors.  It  is  an  evi- 
dent truth  that,  if  a  number  can  be  divided  by  the 
product  of  two  others,  it  can  be  divided  by  each  of 
them.  If  a  number  can  be  parted  into  an  exact  num- 
ber of  twelves,  it  can  be  parted  also  into  a  number  of 
sixes,  twos,  or  fours.  It  is  also  true  that,  if  a  number 
can  be  divided  by  any  other  number,  and  the  quotient 
can  then  be  divided  by  a  third  number,  the  original 
number  can  be  divided  by  the  product  of  the  other 
two.  Thus,  144  is  divisible  by  2  ;  the  quotient,  72,  is 
divisible  by  6  ;  and  the  original  number  is  divisible 
by  6  x  2  or  12.  It  is  also  true  that,  if  two  numbers 
are  prime,  their  product  is  divisible  by  no  numbers 
except  themselves.  Thus,  17  X  H  is  divisible  by  no 
numbers  except  17  and  11.  Though  this  is  a  simple 
proposition,  its  proof  is  not  so,  and  cannot  be  given 
to  the  beginner.  From  these  things  it  follows  that 
the  greatest  common  measure  of  two  numbers  (meas- 
ure being  an  old  word  for  divisor)  is  the  product  of  all 
the  prime  factors  which  the  two  possess  in  common. 
For  example,  the  numbers  90  and  100,  which,  when 
reduced  to  their  prime  factors,  are  2x^X3x3  and 
2  X  2  X  5  X  5>  have  the  common  factors  2  and  5,  and 
are  divisible  by  2  X  5,  or  10.  The  quotients  are  3  X  3 
and  2x5,  or  9  and  10,  which  have  no  common  factor 
remaining,  and  2  X  5,  or  10,  is  the  greatest  common 
measure  of  90  and  100.  The  same  may  be  shown  in 
the  case  of  any  other  numbers.  But  the  method  we 


ELEMENTARY  RULES  OF  ARITHMETIC.  27 

have  mentioned  of  resolving  numbers  into  their  prime 
factors,  being  troublesome  to  apply  when  the  num- 
bers are  large,  is  usually  abandoned  for  another.  It 
happens  frequently  that  a  method  simple  in  principle 
is  laborious  in  practice,  and  the  contrary. 

When  one  number  is  divided  by  another,  and  its 
quotient  and  remainder  obtained,  the  dividend  may 
be  recovered  again  by  multiplying  the  quotient  and 
divisor  together,  and  adding  the  remainder  to  the  pro- 
duct. Thus  171  divided  by  27  gives  a  quotient  6  and 
a  remainder  9,  and  171  is  made  by  multiplying  27  by 
6,  and  adding  9  to  the  product.  That  is,  171  = 
27  X  6  -J-  9.  Now,  from  this  equation  it  is  easy  to 
show  that  every  number  which  divides  171  and  27 
also  divides  9,  that  is,  every  common  measure  of  171 
and  27  is  also  a  common  measure  of  27  and  9.  We 
can  also  show  that  27  and  9  have  no  common  meas- 
ures which  are  not  common  to  171  and  27.  Therefore, 
the  common  measures  of  171  and  27  are  those,  and  no 
others,  which  are  common  to  27  and  9 ;  the  greatest 
common  measure  of  each  pair  must,  therefore,  be  the 
same,  that  is,  the  greatest  common  measure  of  a  di- 
visor and  dividend  is  also  the  greatest  common  meas- 
ure of  the  remainder  and  divisor.  Now  take  the  com- 
mon process  for  finding  the  greatest  common  measure 
of  two  numbers ;  for  example,  360  and  420,  which  is 
as  follows,  and  abbreviate  the  words  greatest  common 
measure  into  their  initials  g.  c.  m. : 


28  ON  THE  STUDY  OF  MATHEMATICS. 

360)420(1 
360 

~60)360(6 
360 


From  the  theorem  above  enunciated  it  appears 
that 

g.  c.  m.  of  420  and  360  is  g.  c.  m.  of  60  and  360 ; 
g.  c.  m.  of  60  and  360  is  60  ; 

because  60  divides  both  60  and  360,  and  no  number 
can  have  a  greater  measure  than  itself.  Thus  may  be 
seen  the  reason  of  the  common  rule  for  finding  the 
greatest  common  measure  of  two  numbers. 

Every  number  which  can  be  divided  by  another 
without  remainder  is  called  a  multiple  of  it.  Thus, 
12,  18,  and  42  are  multiples  of  6,  and  the  last  is  a 
common  multiple  of  6  and  7,  because  it  is  divisible  both 
by  6  and  7.  The  only  things  which  it  is  necessary  to 
observe  on  this  subject  are,  (1),  that  the  product  of 
two  numbers  is  a  common  multiple  of  both ;  (2),  that 
when  the  two  numbers  have  a  common  measure  greater 
than  1,  there  is  a  common  multiple  less  than  their 
product;  (3),  that  when  they  have  no  common  meas- 
ure except  1,  the  least  common  multiple  is  their  pro- 
duct. The  first  of  these  is  evident ;  the  second  will 
appear  from  an  example.  Take  10  and  8,  which  have 
the  common  measure  2,  since  the  first  is  2  x  5  and 
the  second  2x4.  The  product  is  2  X  2  X  4  X  5,  but 


ELEMENTARY  RULES  OF  ARITHMETIC.  2Q 

2  X  4  X  ^  is  also  a  common  multiple,  since  it  is  divis- 
ible by  2  x  4,  or  8,  and  by  2  X  5,  or  10.  To  find  this 
common  multiple  we  must,  therefore,  divide  the  pro- 
duct by  the  greatest  common  measure.  The  third 
principle  cannot  be  proved  in  an  elementary  way,  but 
the  student  may  convince  himself  of  it  by  any  number 
of  examples.  He  will  not,  for  instance,  be  able  to 
find  a  common  multiple  of  8  and  7  less  than  8  X  7 
or  56. 


CHAPTER  IV. 

ARITHMETICAL  FRACTIONS. 

"TT7HEN  the  student  has  perfected  himself  in  the 
*  ^  four  rules,  together  with  that  for  finding  the 
greatest  common  measure,  he  should  proceed  at  once 
to  the  subject  of  fractions.  This  part  of  arithmetic  is 
usually  supposed  to  present  extraordinary  difficulties  ; 
whereas,  the  fact  is  that  there  is  nothing  in  fractions 
so  difficult,  either  in  principle  or  practice,  as  the  rule 
for  finding  the  greatest  common  measure.  We  would 
recommend  the  student  not  to  attend  to  the  distinc- 
tions of  proper  and  improper,  pure  or  mixed  fractions, 
etc. ,  as  there  is  no  distinction  whatever  in  the  rules, 
which  are  common  to  all  these  fractions. 

When  one  number,  as  56,  is  to  be  divided  by  an- 
other, as  8,  the  process  is  written  thus  :  -5B6-.  By  this 
we  mean  that  56  is  to  be  divided  into  8  equal  parts, 
and  one  of  these  parts  is  called  the  quotient.  In  this 
case  the  quotient  is  7.  But  it  is  equally  possible 
to  divide  57  into  8  equal  parts  ;  for  example,  we  can 
divide  57  feet  into  8  equal  parts,  but  the  eighth  part 


ARITHMETICAL  FRACTIONS.  3! 


of  57  feet  will  not  be  an  exact  number  of  feet,  since 
57  does  not  contain  an  exact  number  of  eights  ;  a  part 
of  a  foot  will  be  contained  in  the  quotient  $£-,  and  this 
quotient  is  therefore  called  a  fraction,  or  broken  num- 
ber. If  we  divide  57  into  56  and  1,  and  take  the 
eighth  part  of  each  of  these,  whose  sum  will  give  the 
eighth  part  of  the  whole,  the  eighth  of  56  feet  is  7 
feet ;  the  eighth  of  1  foot  is  a  fraction,  which  we  write 
J,  and  -5^  is  7  -j-  -J,  which  is  usually  written  7  J.  Both 
of  these  quantities  $£-,  and  7i,  are  called  fractions  ;  the 
only  difference  is  that,  in  the  second,  that  part  of  the 
quotient  which  is  a  whole  number  is  separated  from 
the  part  which  is  less  than  any  whole  number. 

There  are  two  ways  in  which  a  fraction  may  be 
considered.  Let  us  take,  for  example,  f .  This  means 
that  5  is  to  be  divided  into  8  parts,  and  |  stands  for 
one  of  these  parts.  The  same  length  will  be  obtained 
if  we  divide  1  into  8  parts,  and  take  5  of  them,  or  find 
i  X  5.  To  prove  this  let  each  of  the  lines  drawn  be- 
low represent  J  of  an  inch ;  repeat  J  five  times,  and 
repeat  the  same  line  eight  times. 


In  each  column  is  £th  of  an  inch  repeated  8  times ; 
that  is  one  inch.     There  are,  then,   5  inches  in  all, 


32  ON  THE  STUDY  OF  MATHEMATICS. 

since  there  are  five  columns.  But  since  there  are  8 
lines,  each  line  is  the  eighth  of  5  inches,  or  f,  but 
each  line  is  also  Jth  of  an  inch  repeated  5  times,  or 
J  X  5-  Therefore,  f  =  JX  5  ;  that  is,  in  order  to  find 
|  inches,  we  may  either  divide  five  inches  into  8  parts, 
and  take  one  of  them,  or  divide  one  inch  into  8  parts, 
and  take^z^  of  them.  The  symbol  f  is  made  to  stand 
for  both  these  operations,  since  they  lead  to  the  same 
result. 

The  most  important  property  of  a  fraction  is,  that 
if  both  its  numerator  and  denominator  are  multiplied 
by  the  same  number,  the  value  of  the  fraction  is  not 
altered ;  that  is,  f  is  the  same  as  J|,  or  each  part  is 
the  same  when  we  divide  12  inches  into  20  parts,  as 
when  we  divide  3  inches  into  5  parts.  Again,  we  get 
the  same  length  by  dividing  1  inch  into  20  parts,  and 
taking  12  of  them,  which  we  get  by  dividing  1  inch 
into  5  parts  and  taking  3  of  them.  This  hardly  needs 
demonstration.  Taking  12  out  of  20  is  taking  3  out 
of  5,  since  for  every  3  which  12  contains,  there  is  a  5 
contained  in  20.  Every  fraction,  therefore,  admits  of 
innumerable  alterations  in  its  form,  without  any  altera- 
tion in  its  value.  Thus,  i  =  J  —  ^  =  |  =  T5^,  etc.  ; 

»=A=A=A.  etc- 

On  the  same  principle  it  is  shown  that  the  terms 
of  a  fraction  may  be  divided  by  any  number  without 
any  alteration  of  its  value.  There  will  now  be  no  diffi- 
culty in  reducing  fractions  to  a  common  denomina- 
tor, in  reducing  a  fraction  to  its  lowest  terms  ;  neither 


ARITHMETICAL  FRACTIONS.  33 

in  adding  nor  subtracting  fractions,  for  all  of  which 
the  rules  are  given  in  every  book  of  arithmetic. 

We  now  come  to  a  rule  which  presents  more  pe- 
culiar difficulties  in  point  of  principle  than  any  at 
which  we  have  yet  arrived.  If  we  could  at  once  take 
the  most  general  view  of  numbers,  and  give  the  be- 
ginner the  extended  notions  which  he  may  afterwards 
attain,  the  mathematics  would  present  comparatively 
few  impediments.  But  the  constitution  of  our  minds 
will  not  permit  this.  It  is  by  collecting  facts  and 
principles,  one  by  one,  and  thus  only,  that  we  arrive 
at  what  are  called  general  notions  ;  and  we  afterwards 
make  comparisons  of  the  facts  which  we  have  acquired 
and  discover  analogies  and  resemblances  which,  while 
they  bind  together  the  fabric  of  our  knowledge,  point 
out  methods  of  increasing  its  extent  and  beauty.  In 
the  limited  view  which  we  first  take  of  the  operations 
which  we  are  performing,  the  names  which  we  give 
are  necessarily  confined  and  partial ;  but  when,  after 
additional  study  and  reflection,  we  recur  to  our  former 
notions,  we  soon  discover  processes  so  resembling  one 
another,  and  different  rules  so  linked  together,  that 
we  feel  it  would  destroy  the  symmetry  of  our  language 
if  we  were  to  call  them  by  different  names.  We  are 
then  induced  to  extend  the  meaning  of  our  terms,  so 
as  to  make  two  rules  into  one.  Also,  suppose  that 
when  we  have  discovered  and  applied  a  rule  and  given 
the  process  which  it  teaches  a  particular  name,  we 
find  that  this  process  is  only  a  part  of  one  more  gen- 


34  ON  THE  STUDY  OF  MATHEMATICS. 

cral,  which  applies  to  all  cases  contained  under  the 
first,  and  to  others  besides.  We  have  only  the  alter- 
native of  inventing  a  new  name,  or  of  extending  the 
meaning  of  the  former  one  so  as  to  merge  the  particu- 
lar process  in  the  more  general  one  of  which  it  is  a 
part.  Of  this  we  can  give  an  instance.  We  began 
with  reasoning  upon  simple  numbers,  such  as  1,  2,  3, 
20,  etc.  We  afterwards  divided  these  into  parts,  of 
which  we  took  some  number,  and  which  we  called 
fractions,  such  as  |,  J,  J,  etc.  Now  there  is  no  num- 
ber which  may  not  be  considered  as  a  fraction  in  as 
many  different  ways  as  we  please.  Thus  7  is  -^  or 
-^-,  etc. ;  12  is  -2^,  -7^,  etc.  Our  new  notion  of  frac- 
tion is,  then,  one  which  includes  all  our  former  ideas 
of  number,  and  others  besides.  It  is  then  customary 
to  represent  by  the  word  number,  not  only  our  first 
notion  of  it,  but  also  the  extended  one,  of  which  the 
first  is  only  a  part.  Those  to  which  our  first  notions 
applied  we  call  whole  numbers,  the  others  fractional 
numbers,  but  still  the  name  number  is  applied  to  both 
2  and  ^,  3  and  j.  The  rule  of  which  we  have  spoken 
is  another  instance.  It  is  called  the  multiplication  of 
fractional  numbers.  Now,  if  we  return  to  our  mean- 
ing of  the  word  multiplication,  we  shall  find  that  the 
multiplication  of  one  fraction  by  another  appears  an 
absurdity.  We  multiply  a  number  by  taking  it  several 
times  and  adding  these  together.  What,  then,  is 
meant  by  multiplying  by  a  fraction?  Still,  a  rule  has 
been  found  which,  in  applying  mathematics,  it  is  ne- 


ARITHMETICAL  FRACTIONS. 


35 


cessary  to  use  for  fractions,  in  all  cases  where  multi- 
plication would  have  been  used  had  they  been  whole 
numbers.  Of  this  we  shall  now  give  a  simple  exam- 
ple. Take  an  oblong  figure  (which  is  called  a  rect- 
angle in  geometry),  such  as  A  BCD,  and  find  the  mag- 
nitudes of  the  sides  AB  and  BC  in  inches.  Draw  the 


B 


A  D 

line  EF  equal  in  length  to  one  inch,  and  the  square 
G,  each  of  whose  sides  is  one  inch.  If  the  lines  AB 
and  BC  contain  an  exact  number  of  inches,  the  rect- 
angle ABCD  contains  an  exact  number  of  squares, 


each  equal  to  G,  and  the  number  of  squares  contained 
is  found  by  multiplying  the  number  of  inches  in  AB 
by  the  number  of  inches  in  BC.  In  the  present  case 
the  number  of  squares  is  3  X  4,  or  12.  Now,  suppose 
another  rectangle  A'B'C'JD',  of  which  neither  of  the 
sides  is  an  exact  number  of  inches  ;  suppose,  for  exam- 
ple, that  A'B'  is  |  of  an  inch,  and  that  B '  C  is  f  of  an 
B' C 


D' 


36  ON  THE  STUDY  OF  MATHEMATICS. 

inch.  We  may  show,  by  reasoning,  that  we  can  find 
how  much  A'JS'C'D'  is  of  G  by  forming  a  fraction 
which  has  the  product  of  the  numerators  of  f  and  f 
for  its  numerator,  and  the  product  of  their  denomina- 
tors for  its  denominator;  that  is,  that  A'B'C'D'  con- 
tains ^  of  G.  Here  then  appears  a  connexion  be- 
tween the  multiplication  of  whole  numbers,  and  the 
formation  of  a  fraction  whose  numerator  is  the  pro- 
duct of  two  numerators,  and  its  denominator  the  pro- 
duct of  the  corresponding  denominators.  These  ope- 
rations will  always  come  together,  that  is  whenever  a 
question  occurs  in  which,  when  whole  numbers  are 
given,  those  numbers  are  to  be  multiplied  together; 
when  fractional  numbers  are  given,  it  will  be  neces- 
sary, in  the  same  case,  to  multiply  the  numerator  by 
the  numerator,  and  the  denominator  by  the  denomina- 
tor, and  form  the  result  into  a  fraction,  as  above. 

This  would  lead  us  to  suspect  some  connexion  be- 
tween these  two  operations,  and  we  shall  accordingly 
find  that  when  whole  numbers  are  formed  into  frac- 
tions, they  may  be  multiplied  together  by  this  very 
rule.  Take,  for  example,  the  numbers  3  and  4,  whose 
product  is  12.  The  first  may  be  written  as  -V5-,  and 
the  second  as  f .  Form  a  fraction  from  the  product 
of  the  numerators  and  denominators  of  these,  which 
will  be  \2_0-,  which  is  12,  the  product  of  3  and  4. 

From  these  considerations  it  is  customary  to  call 
the  fraction  which  is  produced  from  two  others  in  the 
manner  above  stated,  the  product  of  those  two  frac- 


ARITHMETICAL  FRACTIONS.  37 

tions,  and  the  process  of  finding  the  third  fraction, 
multiplication.  We  shall  always  find  the  first  meaning 
of  the  word  multiplication  included  in  the  second,  in 
all  cases  in  which  the  quantities  represented  as  frac- 
tions are  really  whole  numbers.  The  mathematics  are 
not  the  only  branches  of  knowledge  in  which  it  is  cus- 
tomary to  extend  the  meaning  of  established  terms. 
Whenever  we  pass  from  that  which  is  simple  to  that 
which  is  complex,  we  shall  see  the  necessity  of  carry- 
ing our  terms  with  us  and  enlarging  their  meaning, 
as  we  enlarge  our  own  ideas.  This  is  the  only  method 
of  forming  a  language  which  shall  approach  in  any 
degree  towards  perfection  ;  and  more  depends  upon 
a  well- constructed  language  in  mathematics  than  in 
anything  else.  It  is  not  that  an  imperfect  language 
would  deprive  us  of  the  means  of  demonstration,  or 
cramp  the  powers  of  reasoning.  The  propositions  of 
Euclid  upon  numbers  are  as  rationally  established  as 
any  others,  although  his  terms  are  deficient  in  analogy, 
and  his  notation  infinitely  inferior  to  that  which  we 
use.  It  is  the  progress  of  discovery  which  is  checked 
by  terms  constructed  so  as  to  conceal  resemblances 
which  exist,  and  to  prevent  one  result  from  pointing 
out  another.  The  higher  branches  of  mathematics 
date  the  progress  which  they  have  made  in  the  last 
century  and  a  half,  from  the  time  when  the  genius  of 
Newton,  Leibnitz,  Descartes,  and  Hariot  turned  the 
attention  of  the  scientific  world  to  the  imperfect  mech- 
anism of  the  science.  A  slight  and  almost  casual  im- 


38  ON  THE  STUDY  OF  MATHEMATICS. 

provement,  made  by  Hariot  in  algebraical  language, 
has  been  the  foundation  of  most  important  branches 
of  the  science.*  The  subject  of  the  last  articles  is  of 
very  great  importance,  and  will  often  recur  to  us  in 
explaining  the  difficulties  of  algebraical  notation. 

The  multiplication  of  f  by  f  is  equivalent  to  divid- 
ing f  into  2  parts,  and  taking  three  such  parts.  Be- 
cause f  being  the  same  as  -if,  or  1  divided  into  12 
parts  and  10  of  them  taken,  the  half  of  -i§  is  5  of  those 
parts,  or  f-%.  Three  times  this  quantity  will  be  15  of 
those  parts,  or  ^J,  which  is  by  our  rule  the  same  as 
what  we  have  called,  f  multiplied  by  f .  But  the  same 
result  arises  from  multiplying  J  by  f,  or  dividing  | 
into  6  parts  and  taking  5  of  them.  Therefore,  we  find 
that  |  multiplied  by  f  is  the  same  as  -|  multiplied  by 
f,  or  |  X  f  —  f  X  f-  This  proposition  is  usually  con- 
sidered as  requiring  no  proof,  because  it  is  received 
very  early  on  the  authority  of  a  rule  in  the  elements 
of  arithmetic.  But  it  is  not  self-evident,  for  the  truth 
of  which  we  appeal  to  the  beginner  himself,  and  ask 
him  whether  he  would  have  seen  at  once  that  |-  of  an 
apple  divided  into  2  parts  and  3  of  them  taken,  is  the 
same  as  f  of  an  apple,  or  one  apple  and  a-half  divided 
into  six  parts  and  5  of  them  taken. 

An  extension  of  the  same  sort  is  made  of  the  term 
division.  In  dividing  one  whole  number  by  another, 


*The  mathematician  will  be  aware  that  I  allude  to  writing  an  equation 
in  the  form 

x%-\-ax  —  b  =  o;  instead  of 


ARITHMETICAL  FRACTIONS.  39 

for  example,  12  by  2,  we  endeavor  to  find  how  many 
twos  must  be  added  together  to  make  12.  In  passing 
from  a  problem  which  contains  these  whole  numbers 
to  one  which  contains  fractional  quantities,  for  exam- 
ple J  and  J,  it  will  be  observed  that  in  place  of  find- 
ing how  many  twos  make  12,  we  shall  have  to  find 
into  how  many  parts  f  must  be  divided,  and  how  many 
of  them  must  be  taken,  so  as  to  give  |.  If  we  reduce 
these  fractions  to  a  common  denominator,  in  which 
case  they  will  be  JJ-  and  ^ ;  and  if  we  divide  the  sec- 
ond into  8  equal  parts,  each  of  which  will  be  ^L,  and 
take  15  of  these  parts,  we  shall  get  |f,  or  f.  The 
fraction  whose  numerator  is  15,  and  whose  denomina- 
tor is  8,  or  Jg5-,  will  in  these  problems  take  the  place 
of  the  quotient  of  the  two  whole  numbers.  In  the 
same  manner  as  before,  it  may  be  shown  that  this  pro- 
cess is  equivalent  to  the  division  of  one  whole  number 
by  another,  whenever  the  fractions  are  really  whole 
numbers  ;  for  example,  3  is  J-¥2-,  and  15  is  $£-.  If  this 
process  be  applied  to  -32°-  and  ±£-,  the  result  is  J^0-, 
which  is  5,  or  the  same  as  15  divided  by  3.  This  pro- 
cess is  then,  by  extension,  called  division  :  -ig5-  is  called 
the  quotient  of  f  divided  by  f ,  and  is  found  by  multi- 
plying the  numerator  of  the  first  by  the  denominator 
of  the  second  for  the  numerator  of  the  result,  and  the 
denominator  of  the  first  by  the  numerator  of  the  sec- 
ond for  the  denominator  of  the  result.  That  this  pro- 
cess does  give  the  same  result  as  ordinary  division  in 
all  cases  where  ordinary  division  is  applicable,  we  can 


40  ON  THE  STUDY  OF  MATHEMATICS. 

easily  show  from  any  two  whole  numbers,  for  exam- 
ple, 12  and  2,  whose  quotient  is  6.  Now  12  is  -\6-,  and 
2  is  -^,  and  the  rule  for  what  we  have  called  division 
of  fractions  will  give  as  the  quotient  ±£fi,  which  is  6. 
In  all  fractional  investigations,  when  the  beginner 
meets  with  a  difficulty,  he  should  accustom  himself  to 
leave  the  notation  of  fractions,  and  betake  himself  to 
their  original  definition.  He  should  recollect  that  f 
is  1  divided  into  6  parts  and  five  of  them  taken,  or  the 
sixth  part  of  5,  and  he  should  reason  upon  these  sup- 
positions, neglecting  all  rules  until  he  has  established 
them  in  his  own  mind  by  reflexion  on  particular  in- 
stances. These  instances  should  not  contain  large 
numbers,  and  it  will  perhaps  assist  him  if  he  reasons 
on  some  given  unit,  for  example  a  foot.  Let  AB  be 
one  foot,  and  divide  it  into  any  number  of  equal  parts 
(7  for  example)  by  the  points  C,  D,  E,  F,  G,  and  H. 


I  I  I  I  ! 

C  D  E  F  G 


He  must  then  recollect  that  each  of  these  parts  is  ^ 
of  a  foot ;  that  any  two  of  them  together  are  ^  of  a 
foot ;  any  3,  ^,  and  so  on.  He  should  then  accustom 
himself,  without  a  rule,  to  solve  such  questions  as  the 
following,  by  observation  of  the  figure,  dividing  each 
part  into  several  equal  parts,  if  necessary ;  and  he 
may  be  well  assured  that  he  does  not  understand  the 
nature  of  fractions  until  such  questions  are  easy  to 
him. 


ARITHMETICAL  FRACTIONS.  41 

What  is  J  of  f  of  a  foot?  What  is  f  of  |  of  f  of  a 
foot?  Into  how  many  parts  must  ^  of  a  foot  be  di- 
vided, and  how  many  of  them  must  be  taken  to  pro- 
duce ••  of  a  foot?  What  is  -f-  °^  a  f°ot ?  an^  so  on- 


CHAPTER  V. 

DECIMAL  FRACTIONS. 

IT  is  a  disadvantage  attending  rules  received  without 
a  knowledge  of  principles,  that  a  mere  difference 
of  language  is  enough  to  create  a  notion  in  the  mind 
of  a  student  that  he  is  upon  a  totally  different  subject. 
Very  few  beginners  see  that  in  following  the  rule 
usually  called  practice,  they  are  working  the  same 
questions  as  were  proposed  in  compound  multiplica- 
tion ; — that  the  rule  of  three  is  only  an  application  of 
the  doctrine  of  fractions  ;  that  the  rules  known  by  the 
name  of  commission,  brokerage,  interest,  etc.,  are  the 
same,  and  so  on.  No  instance,  however,  is  more  con- 
spicuous than  that  of  decimal  fractions,  which  are 
made  to  form  a  branch  of  arithmetic  as  distinct  from 
ordinary  or  vulgar  fractions  as  any  two  parts  of  the 
subject  whatever.  Nevertheless,  there  is  no  single 
rule  in  the  one  which  is  not  substantially  the  same  as 
the  rule  corresponding  in  the  other,  the  difference 
consisting  altogether  in  a  different  way  of  writing  the 
fractions.  The  beginner  will  observe  that  throughout 


DECIMAL  FRACTIONS.  43 

the  subject  it  is  continually  necessary  to  reduce  frac- 
tions to  a  common  denominator  :  he  will  see,  there- 
fore, the  advantage  of  always  using  either  the  same 
denominator,  or  a  set  of  denominators,  so  closely  con- 
nected as  to  be  very  easily  reducible  to  one  another. 
Now  of  all  numbers  which  c'an  be  chosen  the  most 
easily  manageable  are  10,  100,  1000,  etc.,  which  are 
called  decimal  numbers  on  account  of  their  connexion 
with  the  number  ten.  All  fractions,  such  as  T7^, 
•ffifo,  izff^,  which  have  a  decimal  number  for  the 
denominator,  are  called  decimal  fractions.  Now  a 
denominator  of  this  sort  is  known  whenever  the  num- 
ber of  cyphers  in  it  are  known ;  thus  a  decimal  num- 
ber with  4  cyphers  can  only  be  10,000,  or  ten  thou- 
sand. We  need  not,  therefore,  write  the  denominator, 
provided,  in  its  stead,  we  put  some  mark  upon  the 
numerator,  by  which  we  may  know  the  number  of 
cyphers  in  the  denominator.  This  mark  is  for  our  own 
selection.  The  method  which  is  followed  is  to  point 
off  from  the  numerator  as  many  figures  as  there  are 
cyphers  in  the  denominator.  Thus  ^Vo4  *s  represented 
by  17.334;  fifrfc  thus,  .229.  We  might,  had  we  so 
pleased,  have  represented  them  thus,  173343,  2293 ; 
or  thus,  173343,  2293,  or  in  any  way  by  which  we 
might  choose  to  agree  to  recollect  that  the  denomina- 
tor is  1  followed  by  3  cyphers.  In  the  common  method 
this  difficulty  occurs  immediately.  What  shall  be  done 
when  there  are  not  as  many  figures  in  the  numerator 
as  there  are  cyphers  in  the  denominator?  How  shall 


44  ON  THE  STUDY  OF  MATHEMATICS. 


we  represent  -j^ffa  ?  We  must  here  extend  our  lan- 
guage a  little,  and  imagine  some  method  by  which, 
without  essentially  altering  the  numerator,  it  may  be 
made  to  show  the  number  of  cyphers  in  the  denom- 
inator. Something  of  the  sort  has  already  been  done 
in  representing  a  number  of  tens,  hundreds,  or  thou- 
sands, etc.  ;  for  5  thousands  were  represented  by  5000, 
in  which,  by  the  assistance  of  cyphers,  the  5  is  made 
to  stand  in  the  place  allotted  to  thousands.  If,  in  the 
present  instance,  we  place  cyphers  at  the  beginning  of 
the  numerator,  until  the  number  of  figures  and  cyphers 
together  is  equal  to  the  number  of  cyphers  in  the  de- 
nominator, and  place  a  point  before  the  first  cypher, 
the  fraction  T^8o8^  will  be  represented  thus,  .0088  ;  by 
which  we  understand  a  fraction  wrhose  numerator  is 
88,  and  whose  denominator  is  a  decimal  number  con- 
taining four  cyphers. 

There  is  a  close  connexion  between  the  manner  of 
representing  decimal  fractions,  and  the  decimal  nota- 
tion for  numbers.  Take,  for  example,  the  fraction 
217.3426  or  H^t^F-  You  wil1  recollect  that  2173426 
is  made  up  of  2000000  -f  100000  -f  70000  -f-  3000  -f 
400  +  20  +  6.  If  each  of  these  parts  be  divided  by 
10000,  and  the  quotient  obtained  or  the  fraction  re- 
duced to  its  lowest  terms,  the  result  is  as  follows  : 

2173426  -200  I  10  I  7  I    3       -1      _2_          6 
10000  MO^  100^1000~MOOOO 

We  see,  then,  that  in  the  fraction  217.3426  the  first 
figure  2  counts  two  hundred  ;  the  second  figure,  1, 


DECIMAL  FRACTIONS.  45 

ten,  and  the  third  7  units.  It  appears,  then,  that  all 
figures  on  the  left  of  the  decimal  point  are  reckoned 
as  ordinary  numbers.  But  on  the  right  of  that  point 
we  find  the  figure  3,  which  counts  for  T3^  ;  4,  which 
counts  for  ^j  2,  for  ^^;  and  6,  for  Tof^-  It  ap- 
pears therefore,  that  numbers  on  the  right  of  the  de- 
cimal point  decrease  as  they  move  towards  the  right, 
each  number  being  one-tenth  of  what  it  would  have 
been  had  it  come  one  place  nearer  to  the  decimal 
point.  The  first  figure  on  the  right  hand  of  that  point 
is  so  many  tenths  of  a  unit,  the  second  figure  so  many 
hundredths  of  a  unit,  and  so  on. 

The  learner  should  go  through  the  same  investiga- 
tion with  other  fractions,  and  should  demonstrate  by 
means  of  the  principles  of  fractions,  generally,  such 
exercises  as  the  following,  until  he  is  thoroughly  ac- 
customed to  this  new  method  of  writing  fractions  : 
.68342  =  .6  +  .08  +  .003  +  .0004  +  .00002 
68342          6  8  3  4  2 


__   __ 

'r     ~~  ~*~   +    "*" 


__   __ 

100000  ~~  To  ~  Too  1000  loooo  100000 


.00012  =  .0001  +  .00002  = 


10000    '    100000 

iooo   L6S  1000  =  ~TF  +  Tooo 

16342    .       9 


100      r  1000' 

The  rules  of  addition,  subtraction,  and  multiplica- 
tion may  now  be  understood.  In  addition  and  sub- 
traction, the  keeping  the  decimal  points  under  one 


46  ON  THE  STUDY  OF  MATHEMATICS. 

another  is  equivalent  to  reducing  the  fractions  to  a 
common  denominator,  as  we  may  show  thus :  Take 
two  fractions,  1.5  and  2.125,  or  -J-JJ.  and  f if  g,  which, 
reducing  the  first  to  the  denominator  of  the  second, 
may  be  written  ijgg  and  f if  g.  If  we  add  the  nume- 
rators together,  we  find  the  sum  of  the  fractions  f  §f  g, 

or  3.625 

2125          2.125 
1500          1.5 

3625          3.625 

The  learner  can  now  see  the  connexion  of  the  rule 
given  for  the  addition  of  decimal  fractions  with  that 
for  the  addition  of  vulgar  fractions.  There  is  the 
same  connexion  between  the  rules  of  subtraction.  The 
principle  of  the  rule  of  multiplication  is  as  follows : 
If  two  decimal  numbers  be  multiplied  together,  the 
product  has  as  many  cyphers  as  are  in  both  to- 
gether. Thus  100x1000  =  100000,  10  X  100:=  1000, 
etc.  Therefore  the  denominator  of  the  product,  which 
is  the  product  of  the  denominators,  has  as  many  cy- 
phers as  are  in  the  denominators  of  both  fractions, 
and  since  the  numerator  of  the  product  is  the  product 
of  the  numerators,  the  point  must  be  placed  in  that 
product  so  as  to  cut  off  as  many  decimal  places  as  are 
both  in  the  multiplier  and  the  multiplicand..  Thus: 


24 
)00( 
or  .  004  x  •  06  = .  00024,  etc. 


1000  X  100  ~~~  100000' 


DECIMAL  FRACTIONS.  47 

It  is  a  general  rule,  that  wherever  the  number  of  fig- 
ures falls  short  of  what  we  know  ought  to  be  the  num- 
ber of  decimals,  the  deficiency  is  made  up  by  cyphers. 
It  may  now  be  asked,  whether  all  fractions  can  be 
reduced  to  decimal  fractions?  It  may  be  answered 
that  they  cannot.  It  is  a  principle  which  is  demon- 
strated in  the  science  of  algebra, — that  if  a  number 
be  not  divisible  by  a  prime  number,  no  multiplication 
of  that  number,  by  itself,  will  make  it  so.  Thus  10 
not  being  divisible  by  7,  neither  10  X  10,  nor  10  X  10 
X  10,  etc.,  is  divisible  by  7.  A  consequence  of  this 
is,  that  since  5  and  2  are  the  only  prime  numbers 
which  will  divide  10,  no  fraction  can  be  converted  into 
a  decimal  unless  its  denominator  is  made  up  of  pro- 
ducts, either  of  5  or  2,  or  of  both  combined,  such  as 
5x2,  5x5x2,  5x5x5,  2x2,  etc.  To  show  that 
this  is  the  case,  take  any  fraction  with  such  a  denomi- 
nator ;  for  example,  =• = ^.  Multiply  the  numera- 

o  X  o  X  ° 
tor  and  denominator  by  2,  once  for  every  5,  which  is 

contained  in  the  denominator,  and  the  fraction  will 
then  become 

13X2X2X2  2X2X2X13 


5X5X5X2X2X2'          10x10x10' 

which  is  yVW  or  -104.  In  a  similar  way,  any  fraction 
whose  denominator  has  no  other  factors  than  2  or  5, 
can  be  reduced  to  a  decimal  fraction.  We  first  search 
for  such  a  number  as  will,  when  multiplied  by  the  de- 
nominator, produce  a  decimal  number,  and  then  mul- 


48  ON  THE  STUDY  OF  MATHEMATICS. 

tiply  both  the  numerator  and  denominator  by  that 
number. 

No  fraction  which  has  any  other  factor  in  its  de- 
nominator can  be  reduced  to  a  decimal  fraction  ex- 
actly. But  here  it  must  be  observed  that  in  most 
parts  of  mathematical  computation  a  very  small  error 
is  not  material.  In  different  species  of  calculations, 
more  or  less  exactness  may  be  required ;  but  even  in 
the  most  delicate  operations,  there  is  always  a  limit 
beyond  which  accuracy  is  useless,  because  it  cannot 
be  appreciated.  For  example,  in  measuring  land  for 
sale,  an  error  of  an  inch  in  five  hundred  yards  is  not 
worth  avoiding,  since  even  if  such  an  error  were  com- 
mitted, it  would  not  make  a  difference  which  would 
be  considered  as  of  any  consequence,  as  in  all  prob- 
ability the  expense  of  a  more  accurate  measurement 
would  be  more  than  the  small  quantity  of  land  thereby 
saved  would  be  worth.  But  in  the  measurement  of  a 
line  for  the  commencement  of  a  trigonometrical  sur- 
vey, an  error  of  an  inch  in  five  hundred  yards  would 
be  fatal,  because  the  subsequent  processes  involve 
calculations  of  such  a  nature  that  this  error  would  be 
multiplied,  and  cause  a  considerable  error  in  the  final 
result.  Still,  even  in  this  case,  it  would  be  useless 
to  endeavor  to  avoid  an  error  of  one-thousandth  part 
of  an  inch  in  five  hundred  yards ;  first,  because  no  in- 
struments hitherto  made  would  show  such  an  error: 
and  secondly,  because  if  they  could,  no  material  dif- 


DECIMAL  FRACTIONS.  49 

ference  would  be  made  in  the  result  by  a  correction  of 
it.  Again,  we  know  that  almost  all  bodies  are  length- 
ened in  all  directions  by  heat.  For  example :  A  brass 
ruler  which  is  a  foot  in  length  to-day,  while  it  is  cold, 
will  be  more  than  a  foot  to-morrow  if  it  is  warm.  The 
difference,  nevertheless,  is  scarcely,  if  at  all,  percept- 
ible to  the  naked  eye,  and  it  would  be  absurd  for  a 
carpenter,  in  measuring  a  few  feet  of  mahogany  for  a 
table,  to  attempt  to  take  notice  of  it ;  but  in  the  meas- 
urement of  the  base  of  a  survey,  which  is  several  miles 
in  length  and  takes  many  days  to  perform,  it  is  neces- 
sary to  take  this  variation  into  account,  as  a  want  of 
attention  to  it  may  produce  perceptible  errors  in  the 
result :  nevertheless,  any  error  which  has  not  this  ef- 
fect, it  would  be  useless  to  avoid  even  were  it  pos- 
sible. We  see,  therefore,  that  we  may,  instead  of  a 
fraction,  which  cannot  be  reduced  to  a  decimal,  sub- 
stitute a  decimal  fraction,  if  we  can  find  one  so  near 
to  the  former,  that  the  error  committed  by  the  substi- 
tution will  not  materially  affect  the  result.  We  will 
now  proceed  to  show  how  to  find  a  series  of  decimal 
fractions,  which  approach  nearer  and  nearer  to  a  given 
fraction,  and  also  that,  in  this  approximation,  we  may 
approach  as  near  as  we  please  to  the  given  fraction 
without  ever  being  exactly  able  to  reach  it. 

Take,  for  example,  the  fraction  T7T.  If  we  divide 
the  series  of  numbers  70,  700,  7000,  etc.,  by  11,  we 
shall  obtain  the  following  results  : 


50  ON  THE  STUDY  OF  MATHEMATICS. 

£f  gives  the  quotient  6,  and  the  remainder  4,  and  is  6^ 
TjOfi  "  63  "  7  63T7T 

7000  «  636  <t  4  636^ 

?  QflQ  o  «  6363  "  7         6363T7T 

etc.  etc.  etc. 

Now  observe  that  if  two  numbers  do  not  differ  by 
so  much  as  1,  their  tenth  parts  do  not  differ  by  so 
much  as  -f^,  their  hundredth  parts  by  so  much  as  TJ¥, 
their  thousandth  parts  by  so  much  as  T^o¥>  and  so  on  ; 
and  also  remember  that  T7T  is  the  tenth  part  of  £°-,  the 
hundredth  part  of  -7T°T°-,  and  so  on.  The  two  following 
tables  will  now  be  apparent : 

1Q-     does  not  differ  from  6  by  so  much  as  1 

7_o_o  (t  63  «  i 

™oo  «  636  "  1 

™_op_o.  tt  6363  "  1 

etc.  etc.  etc. 

Therefore 

T7T  does  not  differ  from  -f^  or .  6,  by  so  much  as  y1^  or .  1 

T7T  "  T60%"'63  "  T*IF"-01 

r7T  " 

T7T  " 

etc.  etc.  etc. 

We  have  then  a  series  of  decimal  fractions,  viz.,  .6, 
.63,  .636,  .6363,  .63636,  etc.,  which  continually  ap- 
proach more  and  more  near  to  T7T,  and  therefore  in 
any  calculation  in  which  the  fraction  T7T  appears,  any 
one  of  these  may  be  substituted  for  it,  which  is  suffi- 
ciently near  to  suit  the  purpose  for  which  the  calcula- 
tion is  intended.  For  some  purposes  .636  wrould  be  a 


DECIMAL  FRACTIONS.  5! 

sufficient  approximation;  for  others  .63636363  would 
be  necessary.  Nothing  but  practice  can  show  how 
far  the  approximation  should  be  carried  in  each  case. 

The  division  of  one  decimal  fraction  by  another  is 
performed  as  follows :  Suppose  it  required  to  divide 
6.42  by  1.213.  The  first  of  these  is  f  £J,  an(}  the  sec- 
ond -if ij .  The  quotient  of  these  by  the  ordinary  rule 
is  ffftoo*  or  ffft-  This  fraction  must  now  be  reduced 
to  a  decimal  on  the  principles  of  the  last  article,  by 
the  rule  usually  given,  either  exactly,  or  by  approxi- 
mation, according  to  the  nature  of  the  factors  in  the 
denominator. 

When  the  decimal  fraction  corresponding  to  a  com- 
mon fraction  cannot  be  exactly  found,  it  always  hap- 
pens that  the  series  of  decimals  which  approximates 
to  it,  contains  the  same  number  repeated  again  and 
again.  Thus,  in  the  example  which  we  chose,  T7T  is 
more  and  more  nearly  represented  by  the  fractions  .6, 
.63,  .636,  .6363,  etc.,  and  if  we  carried  the  process  on 
without  end,  we  should  find  a  decimal  fraction  con- 
sisting entirely  of  repetitions  of  the  figures  63  after  the 
decimal  point.  Thus,  in  finding  1,  the  figures  which 
are  repeated  in  the  numerator  are  142857.  This  is 
what  is  commonly  called  a  circulating  decimal,  and 
rules  are  given  in  books  of  arithmetic  for  reducing 
them  to  common  fractions.  We  would  recommend 
to  the  beginner  to  omit  all  notice  of  these  fractions, 
as  they  are  of  no  practical  use,  and  cannot  be  thor- 
oughly understood  without  some  knowledge  of  alge- 


52  ON  THE  STUDY  OF  MATHEMATICS. 

bra.  It  is  sufficient  for  the  student  to  know  that  he 
can  always  either  reduce  a  common  fraction  to  a  deci- 
mal, or  find  a  decimal  near  enough  to  it  for  his  pur- 
pose, though  the  calculation  in  which  he  is  engaged 
requires  a  degree  of  accuracy  which  the  finest  micro- 
scope will  not  appreciate.  But  in  using  approximate 
decimals  there  is  one  remark  of  importance,  the  ne- 
cessity for  which  occurs  continually. 

Suppose  that  the  fraction  2.143876  has  been  ob- 
tained, and  that  it  is  more  than  sufficiently  accurate 
for  the  calculation  in  which  it  is  to  be  employed.  Sup- 
pose that  for  the  object  proposed  it  is  enough  that 
each  quantity  employed  should  be  a  decimal  fraction 
of  three  places  only,  the  quantity  2.143876  is  made  up 
of  2.143,  as  far  as  three  places  of  decimals  are  con- 
cerned, which  at  first  sight  might  appear  to  be  what 
we  ought  to  use,  instead  of  2.143876.  But  this  is  not 
the  number  which  will  in  this  case  give  the  utmost 
accuracy  which  three  places  of  decimals  will  admit 
of;  the  common  usages  of  life  will  guide  us  in  this 
case.  Suppose  a  regiment  consists  of  876  men,  we 
should  express  this  in  what  we  call  round  numbers, 
which  in  this  case  would  be  done  by  saying  how  many 
hundred  men  there  are,  leaving  out  of  consideration 
the  number  76,  which  is  not  so  great  as  100  ;  but  in 
doing  this  we  shall  be  nearer  the  truth  if  we  say  that 
the  regiment  consists  of  900  men  instead  of  800,  be- 
cause 900  is  nearer  to  876  than  800.  In  the  same 
manner,  it  will  be  nearer  the  truth  to  write  2.144  in- 


DECIMAL  FRACTIONS.  53 

stead  of  2.143,  if  we  wish  to  express  2.143876  as  nearly 
as  possible  by  three  places  of  decimals,  since  it  will 
be  found  by  subtraction  that  the  first  of  these  is  nearer 
to  the  third  than  the  second.  Had  the  fraction  been 
2.14326,  it  would  have  been  best  expressed  in  three 
places  by  2.143;  had  it  been  2.1435,  it  would  have 
been  equally  well  expressed  either  by  2.143  or  2.144, 
both  being  equally  near  the  truth;  but  2.14351  is  a 
little  more  nearly  expressed  by  2.144  than  by  2.143. 

We  have  now  gone  through  the  leading  principles 
of  arithmetical  calculation,  considered  as  a  part  of 
general  Mathematics.  With  respect  to  the  commer- 
cial rules,  usually  considered  as  the  grand  object  of 
an  arithmetical  education,  it  is  not  within  the  scope 
of  this  treatise  to  enter  upon  their  consideration.  The 
mathematical  student,  if  he  is  sufficiently  well  versed 
in  their  routine  for  the  purposes  of  common  life,  may 
postpone  their  consideration  until  he  shall  have  be- 
come familiar  with  algebraical  operations,  when  he 
will  find  no  difficulty  in  understanding  the  principles 
or  practice  of  any  of  them.  He  should,  before  com- 
mencing the  study  of  algebra,  carefully  review  what 
he  has  learnt  in  arithmetic,  particularly  the  reasonings 
which  he  has  met  with,  and  the  use  of  the  signs  which 
have  been  introduced.  Algebra  is  at  first  only  arith- 
metic under  another  name,  and  with  more  general 
symbols,  nor  will  any  reasoning  be  presented  to  the 
student  which  he  has  not  already  met  with  in  estab- 
lishing the  rules  of  arithmetic.  His  progress  in  the 


54  ON  THE  STUDY  OF  MATHEMATICS. 

former  science  depends  most  materially,  if  not  alto- 
gether, upon  the  manner  in  which  he  has  attended  to 
the  latter ;  on  which  account  the  detail  into  which  we 
have  entered  on  some  things  which  to  an  intelligent 
person  are  almost  self-evident,  must  not  be  deemed 
superfluous. 

When  the  student  is  well  acquainted  with  the  prin- 
ciples and  practice  of  arithmetic,  and  not  before,  he 
should  commence  the  study  of  algebra.  It  is  usual 
to  begin  algebra  and  geometry  together,  and  if  the 
student  has  sufficient  time,  it  is  the  best  plan  which 
he  can  adopt.  Indeed,  we  see  no  reason  why  the  ele- 
ments of  geometry  should  not  precede  those  of  alge- 
bra, *and  be  studied  together  with  arithmetic.  In  this 
case  the  student  should  read  some  treatise  which  re- 
lates to  geometry,  first.  It  is  hardly  necessary  to  say 
that  though  we  have  adopted  one  particular  order, 
yet  the  student  may  reverse  or  alter  that  order  so  as 
to  suit  the  arrangement  of  his  own  studies. 

We  now  proceed  to  the  first  principles  of  algebra, 
and  the  elucidation  of  the  difficulties  which  are  found 
from  experience  to  be  most  perplexing  to  the  begin- 
ner. We  suppose  him  to  be  well  acquainted  with 
what  has  been  previously  laid  down  in  this  treatise, 
particularly  with  the  meaning  of  the  signs  -|-,  — ,  X, 
and  the  sign  of  division. 


CHAPTER  VI. 

ALGEBRAICAL  NOTATION  AND  PRINCIPLES. 

WHENEVER  any  idea  is  constantly  recurring, 
the  best  thing  which  can  be  done  for  the  per- 
fection of  language,  and  consequent  advancement  of 
knowledge,  is  to  shorten  as  much  as  possible  the  sign 
which  is  used  to  stand  for  that  idea.  All  that  we  have 
accomplished  hitherto  has  been  owing  to  the  short 
and  expressive  language  which  we  have  used  to  rep- 
resent numbers,  and  the  operations  which  are  per- 
formed upon  them.  The  first  step  was  to  write  simple 
signs  for  the  first  numbers,  instead  of  words  at  full 
length,  such  as  8  and  7,  instead  of  eight  and  seven. 
The  next  was  to  give  these  signs  an  additional  mean- 
ing, according  to  the  manner  in  which  they  were  con- 
nected with  one  another;  thus  187  was  made  to  rep- 
resent one  hundred  added  to  eight  tens  added  to  seven. 
The  next  was  to  give  by  new  signs  directions  when  to 
perform  the  operations  of  addition,  subtraction,  mul- 
tiplication, and  division  ;  thus  5  -f-  8  was  made  to  rep- 
resent 8  added  to  5,  and  so  on.  With  these  signs 


56  ON  THE  STUDY  OF  MATHEMATICS. 

reasonings  were  made,  and  truths  discovered  which 
are  common  to  all  numbers  ;  not  at  once  for  every 
number,  but  by  taking  some  example,  by  reasoning 
upon  it,  and  by  producing  a  result ;  this  result  led  to 
a  rule  which  was  declared  to  be  a  rule  which  held 
equally  good  for  all  numbers,  because  the  reasoning 
which  produced  it  might  have  been  applied  to  any 
other  example  as  well  as  to  the  one  which  was  chosen. 
In  this  way  we  produced  some  results,  and  might  have 
produced  many  more  ;  the  following  is  an  instance  : 
half  the  sum  of  two  numbers  added  to  half  their  differ- 
ence, gives  the  greater  of  the  two  numbers.  For  ex- 
ample, take  16  and  10,  half  their  sum  is  13,  half  their 
difference  is  3  ;  if  we  add  13  and  3  we  get  16,  the 
greater  of  the  two  numbers.  We  might  satisfy  our- 
selves of  the  truth  of  this  same  proposition  for  any 
other  numbers,  such  as  27  and  8,  15  and  19,  and  so 
on.  If  we  then  make  use  of  signs,  we  find  the  follow- 
ing truths  : 

16  +  10         16  —  10 
~~2~~  ~2~ 

27  +  8  27—8    _27 

15+9  15-9 


and  so  on.  If,  then,  we  choose  any  two  numbers, 
and  call  them  the  first  and  second  numbers,  and  call 
that  the  first  number  which  is  the  greater  of  the  two, 
we  have  the  following  : 


ALGEBRAICAL  NOTATION  AND  PRINCIPLES.  57 

First  No.  4-  Second  No.       First  No.  —  Second  No. 
~^~  ~2~ 

First  No. 

In  this  way  we  might  express  anything  which  is  true 
of  all  numbers,  by  writing  First  No.,  Second  No.,  etc., 
for  the  different  numbers  which  enter  into  our  propo- 
sition, and  we  might  afterwards  suppose  the  First 
No.,  the  Second  No.,  etc.,  to  be  any  which  we  please. 
In  this  way  we  might  write  down  the  following  asser- 
tion, which  we  should  find  to  be  always  true  : 

(First  No.  -f  Second  No.)  X  (First  No.—  Second  No.) 
=  First  No.  X  First  No.  —  Second  No.  X  Second  No. 

When  any  sentence  expresses  that  two  numbers  or 
collections  of  numbers  are  equal  to  one  another,  it  is 
called  an  equation*  thus  7+  5  =  12  is  an  equation,  and 
the  sentences  written  just  above  are  equations. 

Now  the  next  question  is,  could  we  not  avoid  the 
trouble  of  writing  First  No.,  Second  No.,  etc.,  so  fre- 
quently? This  is  done  by  putting  letters  of  the  alpha- 
bet to  stand  for  these  numbers.  Suppose,  e.  g.,  we 
let  x  stand  for  the  first  number,  and  y  for  the  second, 
the  two  assertions  already  made  will  then  be  written  : 

.  *—y  —  x 

~~~ 


(x  _|-j)x(.*  —  y)  —x  X  x— 

By  the  use  of  letters  we  are  thus  enabled  to  write 

sentences  which  say  something  of  all  numbers,  with  a 

*As  now  usually  defined  an  equation  always  contains  an  unknown  quan- 
tity.    See  also  p.  91.—  Ed, 


58  ON  THE  STUDY  OF  MATHEMATICS. 

very  small  part  only  of  the  time  and  trouble  necessary 
for  writing  the  same  thing  at  full  length.  We  now 
proceed  to  enumerate  the  various  symbols  which  are 
used. 

1.  The  letters  of  the  alphabet  are  used  to  stand 
for  numbers,  and  whenever  a  letter  is  used  it  means 
either  that  any  number  may  be  used  instead  of  that 
letter,  or  that  the  number  which  the  letter  stands  for 
is  not  known,  and  that  the  letter  supplies  its  place  in 
all  the  reasonings  until  it  is  known. 

2.  The  sign  -j-  is  used  for  addition,  as  in  arithme- 
tic.  Thus  x  -\-  z  is  the  sum  of  the  numbers  represented 
by  x  and  z.     The  following  equations  are  sufficiently 
evident  : 


If  a  =  b,  then  a  +  c  =  b  +  c,   a-\-  c  +  d=b  +  c-\-  d, 
etc. 

3.  The  sign  —  is  used  for  subtraction,  as  in  arith- 
metic.    The  following  equations  will  show  its  use  : 

x  -\-  a  —  b  —  c  -\-  e  =  x  -j-  a-\-  e  —  b  —  c 
=  a  —  c-\-  e  —  b-}-  x. 

If  a  =  b,  a  —  c  =  b  —  c,  a  —  c  +  d=b  —  c  -\-  d,  etc. 

4.  The  sign   X    is   used   for  multiplication   as  in 
arithmetic,  but  when  two  numbers  represented  by  let- 
ters are  multiplied  together  it  is  useless,  since  a  X  b 
can  be  represented  by  putting  a  and  b  together  thus, 
ab.     Also  a  X  b  X  c  is  represented  by  a  be;  #  X  #  X  #> 
for  the  present  we  represent  thus,  a  a  a.     When  two 
numbers  are  multiplied  together,  it  is  necessary  to 


ALGEBRAICAL  NOTATION  AND  PRINCIPLES.  59 

keep  the  sign  X  ;  otherwise  7  x  5  or  35  would  be  mis- 
taken for  75.  It  is,  however,  usual  to  place  a  point 
between  two  numbers  which  are  to  be  multiplied  to- 
gether; thus  7x5X3  is  written  7.5.3.  But  this 
point  may  sometimes  be  mistaken  for  the  decimal 
point  :  this  will,  however,  be  avoided  by  always  writ- 
ing the  decimal  point  at  the  head  of  the  figure,  viz., 
by  writing  £f  J£L  thus,  234'  61. 


5.  Division  is  written  as  in   arithmetic  ;    thus,  -=- 

b 

signifies  that  the  number  represented  by  a  is  to  be  di- 
vided by  the  number  represented  by  b. 

6.  All  collections  of  numbers  are  called  expres- 
sions ;    thus,  a-{-b,  a-\-b  —  c,  aa-\-bb  —  d,  are  alge- 
braical expressions. 

7.  When  two  expressions  are  to  be  multiplied  to- 
gether, it  is  indicated  by  placing  them  side  by  side, 
and    inclosing  each    of  them  in  brackets.     Thus,  if 
a  -\-b-\-c  is  to  be  multiplied  by  d-}-e-\-f,  the  product 
is  written  in  any  of  the  following  ways  : 


[</+,+/], 


c.d+e+f, 


8.  That  a  is  greater  than  b  is  written  thus,  0>  b. 

9.  That  a  is  less  than  b  is  written  thus,  a<^b, 

10.  When  there  is  a  product  in  which  all  the  fac- 
tors are  the  same,  such  as  xxxxx,  which  means  that 


60  ON  THE  STUDY  OF  MATHEMATICS. 

five  equal  numbers,  each  of  which  is  represented  by 
x,  are  multiplied  together,  the  letter  is  only  written 
once,  and  above  it  is  written  the  number  of  times 
which  it  occurs,  thus  xxxxx  is  written  x5.  The  fol- 
lowing table  should  be  carefully  studied  by  the  stu- 
dent: 

x  X  x  or  xx  is  written  x2, 

and  is  called  the  square,  or  second  power  of  x. 

xy^xXxorxxx  is  written  x9, 

and  is  called  the  cube  or  third  power  of  x. 

xX^X^X^orxxxxis  written  x*, 
and  is  called  the  fourth  power  of  x. 

.^X^X-^X^X^  or  xxxxx  is  written  x5, 
and  is  called  the  fifth  power  of  x, 

etc.,  etc.,  etc. 

There  is  no  point  which  is  so  likely  to  create  con- 
fusion in  the  ideas  of  a  beginner  as  the  likeness  be- 
tween such  expressions  as  4x  and  x*.  On  this  account 
it  would  be  better  for  him  to  omit  using  the  latter  ex- 
pression, and  to  put  xxxx  in  its  place  until  he  has 
acquired  some  little  facility  in  the  operations  of  alge- 
bra. If  he  does  not  pursue  this  course,  he  must  re- 
collect that  the  4,  in  these  two  expressions,  has  differ- 
ent names  and  meanings.  In  4x  it  is  called  a  coeffi- 
cient, in  x*  an  exponent  or  index. 

The  difference  of  meaning  will  be  apparent  from 
the  following  tables : 


ALGEBRAICAL  NOTATION  AND  PRINCIPLES.  6l 

2x  =  x  --  x  x*  =  xx  =  xx 


x  =  x--  x--  x  x=xy^xy^x 

4x  =  x-\-x-}-x-\-x,          x*  =  xX  x  X  x  X-*  or  xxxx, 
etc.,  etc. 

If*  =  3  2x=  6  x*=  9, 
3*:=  9  x*  =  2T, 
4x  =  l2  *4  =  81, 

The  beginner  should  frequently  write  for  himself 
such  expressions  as  the  following  : 


=  aaa  ax  -\-aaaax-\-  a  a  a  ax  -\-aaaax  -\-aaaax. 


aa  —  bb       aa  —  bb' 
—  £3        aaa 


a2—&2          aa  —  bb  a  +  b 

With  many  such  expressions  every  book  on  algebra 
will  furnish  him,  and  he  should  then  satisfy  himself 
of  their  truth  by  putting  some  numbers  at  pleasure 
instead  of  the  letters,  and  making  the  results  agree 
with  one  another.  Thus,  to  try  the  expression 

0: 


a  —  b    ~ 

or,  which  is  the  same, 
aaa — bbb 


=  aa-\-  ab  -\-  bb. 


a  —  b 

Let  a  stand  for  6  and  b  stand  for  4,  then,  if  this  ex- 
pression be  true, 


62  ON  THE  STUDY  OF  MATHEMATICS. 


which  is  correct,  since  each  of  these  expressions  is 
found,  by  calculation,  to  be  76. 

The  student  should  then  exercise  himself  in  the 
solution  of  such  questions  as  the  following  :  What  is 

«*  +  *  --  '*    +  -a  '--a, 
a  -f-  b        a  —  b 

I.  when  a  stands  for  6,  and  b  for  5,  II.  when  a  stands 
for  13,  and  b  for  2,  and  so  on.  He  should  stop  here 
until  he  has,  by  these  means,  made  the  signs  familiar 
to  his  eye  and  their  meaning  to  his  mind  ;  nor  should 
he  proceed  to  any  further  algebraical  operations  until 
he  can  readily  find  the  value  of  any  algebraical  ex- 
pression when  he  knows  the  numbers  which  the  letters 
stand  for.  He  cannot,  at  this*  period  of  his  course, 
write  too  many  algebraical  expressions,  and  he  must 
particularly  avoid  slurring  over  the  sense  of  what  he 
has  before  him,  and  must  write  and  rewrite  each  ex- 
pression until  the  meaning  of  the  several  parts  forces 
itself  upon  his  memory  at  first  sight,  without  even 
the  necessity  of  putting  it  in  words.  It  is  the  neglect- 
ing to  do  this  which  renders  the  operations  of  algebra 
so  tedious  to  the  beginner.  He  usually  proceeds  to 
the  addition,  subtraction,  etc.,  of  symbols,  of  the 
meaning  of  which  he  has  but  an  imperfect  idea,  and 
which  have  been  newly  introduced  to  him  in  such 
numbers  that  perpetual  confusion  is  the  consequence. 
We  cannot,  therefore,  use  too  many  arguments  to  in- 


ALGEBRAICAL  NOTATION  AND  PRINCIPLES.  63 

duce  him  not  to  mind  the  drudgery  of  reducing  alge- 
braical expressions  into  figures.  This  is  the  connect- 
ing link  between  the  new  science  and  arithmetic,  and, 
unless  that  link  be  well  fastened,  the  knowledge  which 
he  has  previously  acquired  in  arithmetic  will  help  him 
but  little  in  acquiring  algebra. 

The  order  of  the  terms  of  any  algebraical  expres- 
sion may  be  changed  without  changing  the  value  of 
that  expression.  This  needs  no  proof,  and  the  follow- 
ing are  examples  of  the  change : 

a -\- b  -\-  a  b  -\-  c -\-  ac  —  d — e — de — f= 
a  —  </-(-  b  —  e-\-  ab  —  de-}-  c — f-\-  ac  = 
a  -\-  b  —  d —  e  —  de  — f-\-  a  c  -f-  c  -f-  a  b  = 
ab  A-  ac  —  d 'e  -{-  a  -f-  b  -f-  c  —  e  — f —  d. 
When  the  first  term  changes  its  place,  as  in  the  fourth 
of  these  expressions,  the  sign  -|-  is  put  before  it,  and 
should,  properly  speaking,  be  written  wherever  there 
is  no  sign,  to  indicate  that  the  term  in  question  in- 
creases the  result  of  the  rest,  but  it  is  usually  omitted. 
The  negative  sign  is  often  written  before  the  first 
term,  as  in  the  expression  — a-\-b:  but  it  must  be  re- 
collected that  this  is  written  on  the  supposition  that 
a  is  subtracted  from  what  comes  after  it. 

When  an  expression  is  written  in  brackets,  with 
some  sign  before  it,  such  as  a  —  (b  —  <r),  it  is  under- 
stood that  the  expression  in  brackets  is  to  be  consid- 
ered as  one  quantity,  and  that  its  result  or  total  is  to 
be  connected  with  the  rest  by  the  sign  which  precedes 
the  brackets.  In  this  example  it  is  the  difference  of  b 


64  ON  THE  STUDY  OF  MATHEMATICS. 

and  c  which  is  to  be  subtracted  from  a.  If  0  =  12, 
^  =  6,  and  c  =  ±,  this  is  10.  In  the  expression  a  —  b 
made  by  subtracting  b  from  a,  too  much  has  been  sub- 
tracted by  the  quantity  c,  since  it  is  not  b,  but  b  —  c, 
which  must  be  subtracted  from  a.  In  order,  therefore, 
to  make  a  —  (^  —  t),  c  must  be  added  to  a  —  b,  which 
gives  a  —  b-{-c.  Therefore,  a  —  (^  —  c~}  =  a  —  b-\-c. 
Similarly 

—  (c  +  d—  e—  f}  =  a  +  b  —  c  —  d-\-  e+f, 


When  the  positive  sign  is  written  before  an  ex- 
pression in  brackets,  the  brackets  may  be  omitted 
altogether,  unless  they  serve  to  show  that  the  expres- 
sion in  question  is  multiplied  by  some  other.  Thus, 
instead  of  (*  +  £  +  *)•+;&/+*+/),  we  may  write 
a  -{-  b  -j-  c  -j-  d-\-  e  -\-f,  which  is,  in  fact,  only  saying 
that  two  wholes  may  be  added  together  by  adding  to- 
gether all  the  parts  of  which  they  are  composed.  But 
the  expression  a-{-(b-\-c)  (d-\-  e)  must  not  be  written 
thus:  a  -\-  b  -{-  c  (d  -\-  e)  ,  since  the  first  expresses  that 
(£-f  c)  must  be  multiplied  by  (d-\-  e)  and  the  product 
added  to  a,  and  the  second  that  c  must  be  multiplied 
by  (d-\-  e)  and  the  product  added  to  a-\-  b.  If  a,  b,  c, 
d,  and  e,  stand  for  1,  2,  3,  4,  and  5,  the  first  is  46  and 
the  second  30. 

When  two  or  more  quantities  have  exactly  the 
same  letters  repeated  the  same  number  of  times,  such 
as  402^3,  and  602^3,  they  may  be  reduced  into  one  by 


ALGEBRAICAL  NOTATION  AND  PRINCIPLES.  65 

merely  adding  the  coefficients  and  retaining  the  same 
letters.  Thus,  2 a +  3 a  is  50,  §bc — kbc  is  2 be, 
%(x+y)-\-%(x-\-y)  is  5(>+_y).  These  things  are 
evident,  but  beginners  are  very  liable  to  carry  this 
farther  than  they  ought,  and  to  attempt  to  reduce  ex- 
pressions which  do  not  admit  of  reduction.  For  ex- 
ample, they  will  say  that  3/£-f~^2  is  kb  or  4^2,  neither 
of  which  is  true,  except  when  b  stands  for  1.  The  ex- 
pression 3^-f-^2,  or  %b-\-bb,  cannot  be  made  more 
simple  until  we  know  what  b  stands  for.  The  follow- 
ing table  will,  perhaps,  be  of  service  : 

is  not  9a5^5 
is  not  2a 
2ba-\-Sb          is  not  50  b. 

Such  are  the  mistakes  which  beginners  almost  uni- 
versally make,  mostly  for  want  of  a  moment's  consid- 
eration. They  attempt  to  reduce  quantities  which 
cannot  be  reduced,  which  they  do  by  adding  the  ex- 
ponents of  letters  as  well  as  their  coefficients,  or  by 
collecting  several  terms  into  one,  and  leaving  out  the 
signs  of  addition  and  subtraction.  The  beginner  can- 
not too  often  repeat  to  himself  that  two  terms  can 
never  be  made  into  one,  unless  both  have  the  same 
letters,  each  letter  being  repeated  the  same  number 
of  times  in  both,  that  is,  having  the  same  index  in 
both.  When  this  is  the  case,  the  expressions  may  be 
reduced  by  adding  or  subtracting  the  coefficients  ac- 
cording to  the  sign,  and  affixing  the  common  letters 
with  their  indices.  When  there  is  no  coefficient,  as 


66  ON  THE  STUDY  OF  MATHEMATICS. 

in  the  expression  cPb,  the  quantity  represented  by  a2  b 
being   only  taken    once,    1   is  called   the  coefficient. 

Thus, 

—  ab 


The  student  must  also  recollect  that  he  is  not  at  lib- 
erty to  change  an  index  from  one  letter  to  another,  as 
by  so  doing  he  changes  the  quantity  represented. 
Thus  cfib  and  aft  are  quantities  totally  distinct,  the 
first  representing  aaaab  and  the  second  abbbb.  The 
difference  in  all  the  cases  which  we  have  mentioned 
will  be  made  more  clear,  by  placing  numbers  at  pleas- 
ure instead  of  letters  in  the  expressions,  and  calculat- 
ing their  values  ;  but,  in  conclusion,  the  following  re- 
mark must  be  attended  to.  If  it  were  asserted  that  the 


.  .     , 

expression—       —is  the  same  as  04-  b  —  ^—  —,,  and 

a  -\-  b  La  —  b 

we  wish  to  proceed  to  see  whether  this  is  always  the 
case  or  no,  if  we  commence  accidentally  by  supposing 
b  to  stand  for  2  and  a  for  4,  we  shall  find  that  the  first 
is  the  same  as  the  second,  each  being  3J.  But  we 
must  not  conclude  from  this  that  they  are  always  the 
same,  at  least  until  we  have  tried  whether  they  are  so, 
when  other  numbers  are  substituted  for  a  and  b.  If 
we  place  6  and  8  instead  of  a  and  b,  we  shall  find  that 
the  two  expressions  are  not  equal,  and  therefore  we 
must  conclude  that  they  are  not  always  the  same. 
Thus  in  the  expressions  3*  —  4  and  2x-{-  8,  if  x  stand 
for  12,  these  are  the  same,  but  if  it  stands  for  any 
other  number  they  are  not  the  same. 


CHAPTER  VII. 

ELEMENTARY  RULES  OF  ALGEBRA. 

'T^HE  student  should  be  very  well  acquainted  with 
-**  the  principles  and  notation  hitherto  laid  down 
before  he  proceeds  to  the  algebraical  rules  for  addi- 
tion and  subtraction.  He  should  then  take  some  sim- 
ple examples  of  each,  and  proceed  to  find  the  sum 
and  difference  by  reasoning  as  follows.  Suppose  it  is 
required  to  add  c — d  to  a  —  b.  The  direction  to  do 
this  may  either  be  written  in  the  common  way  thus  : 

a  —  b 
c  —  d 

Add" 
or  more  properly  thus  :  Find  (a  —  V)-\-(c  —  d}. 

If  we  add  c  to  a,  or  find  a  -f-  c,  we  have  too  much  : 
first,  because  it  is  not  a  which  is  to  be  increased  by 
c  —  d  but  a  —  b  ;  this  quantity  must  therefore  be  de- 
creased by  b  on  this  account,  or  must  become  a-\-c — b  ; 
but  this  is  still  too  great,  because  it  is  not  c  which  was 
to  be  added  but  c  —  d\  it  must  therefore  be  decreased 
by  d  on  this  account,  or  must  become  a  -|-  c —  b — d  or 


68  ON  THE  STUDY  OF  MATHEMATICS. 

a  —  b-{-  c  —  d.  From  a  few  reasonings  of  this  sort  the 
rule  may  be  deduced  ;  and  not  till  then  should  an  ex- 
ample be  chosen  so  complicated  as  to  make  the  stu- 
dent lose  sight  for  one  moment  of  his  demonstration. 
The  process  of  subtraction  we  have  already  performed. 
and  from  a  few  examples  of  this  method  the  rule  may 
be  deduced. 

The  multiplication  of  a  by  c  —  d  is  performed  thus  : 
a  is  to  be  taken  c  —  d  times.  Take  it  first  c  times  or 
find  ac.  This  is  too  great,  because  a  has  been  taken 
too  many  times  by  d.  From  ac  we  must  therefore 
subtract  d  times  a,  or  ad,  and  the  result  is  that 

a{c  —  d}  =  ac  —  ad. 

This  may  be  verified  from  arithmetic,  in  which  the 
same  process  is  shown  to  be  correct  ;  and  this  whether 
the  numbers  a,  c,  and  d  are  whole  or  fractional.  For 
example,  it  will  be  found  that  6(14  —  9)  or  6y5  is 
the  same  as  6  X  14  —  6  X  9,  or  as  84  —  54.  Also  that 

*(*  —  &)>  or  *XT**  is  the  same  as  fX}—  f  X  T2i>> 
or  as  /T  —  ¥4F.  Upon  similar  reasoning  the  following 
equations  may  be  proved  : 

#(£-j-  c  —  d}  =  ab  -f-  ac  —  ad. 
(J>-\-pq  —  ar)xz=pxz  -\-pqxz  —  arxz. 
,  or  (aa 


Also  when  a  multiplication  has  been  performed,  the 
process  may  be  reversed  and  the  factors  of  it  may  be 
given.  Thus,  on  observing  the  expression 


ELEMENTARY  RULES  OF  ALGEBRA.  69 

ab  —  ac-\-  #2, 
or  ab  —  ac-\-aa, 

we  see  that  in  its  formation  every  term  has  been  mul- 
tiplied by  a  ;  that  is,  it  has  been  made  by  multiplying 

b  —  c  -|-  a  by  a, 
or  a  by  b  —  c-\-a. 
There  will  now  be  no  difficulty  in  perceiving  that 


a(a  — 

It  is  proved  in  arithmetic  that  if  numbers,  whether 
whole  or  fractional,  are  multiplied  together,  the  pro- 
duct remains  the  same  when  the  order  in  which  they 
are  multiplied  is  changed.  Thus  6x4x3  =  3x6x4  = 
4x^x3,  etc.,  and  |  xf  =  |X|,  etc.  Also,  that  a 
part  of  the  multiplication  may  be  made,  and  the  par- 
tial product  substituted  instead  of  the  factors  which 
produced  it,  thus,  3x4X^X6  is  12X&X6,  or  15x4x6, 
or  90x4.  From  these  rules  two  complicated  single 
terms  may  be  multiplied  together,  and  the  product 
represented  in  the  most  simple  manner  which  the  case 
admits  of.  Thus  if  it  be  required  to  multiply 

6tf3^4*:,  which  is  §aaabbbbc 
by  12a2^8^//,  which  is  \Zaabbbcccd, 

the  product  is  written  thus  : 

§aaabbbbc  YZaabbbccc  d, 


70  ON  THE  STUDY  OF  MATHEMATICS. 

which  multiplication  may  be  performed  in  the  follow- 
ing order 

SyyZaaaaabbbbbbbccccd, 

which  is  represented  by  72  a5  ft1  c*  d.  A  few  examples 
of  this  sort  will  establish  the  rule  for  the  multiplica- 
tion of  such  quantities  which  is  usually  given  in  the 
treatises  on  Algebra. 

It  is  to  be  recollected  that  in  every  algebraical 
formula  which  is  true  of  all  numbers,  any  algebraical 
expression  may  be  substituted  for  one  of  the  letters, 
provided  care  is  taken  to  make  the  substitution  wher- 
ever that  letter  occurs.  Thus  from  the  formula  : 


we  may  deduce  the  following  by  making  substitutions 
for  a.  If  this  formula  be  always  true,  it  is  true  when 
a  is  equal  to  p-\-g,  that  is,  it  is  true  if  p-\-q  be  put 
instead  of  a  wherever  that  letter  occurs  in  the  form- 
ula. Therefore, 


Similarly,  (b  -f  mf  —  P  =  (26  +  m)mt 


=  ±xy,  and  so  on. 

We  have  already  established  the  formula, 
(/  —  q)a  =  ap  —  aq. 

Instead  of  a  let  us  put  r  —  s,  and  this  formula  be- 
comes 

(/  —  ?)(>—  s)  =  (r  —  s)p  —  (r—.  f)?. 


ELEMENTARY  RULES  OF  ALGEBRA.          71 

But 

(r  —  s)p  =  pr  —  ps,    and   (r  —  s^q  =  qr  —  qs. 
Therefore 

(/  —  ?)O  —  s^^pr  —  ps—  (qr—  qs} 
=pr—ps  —  qr-\-qs. 

By  reasoning  in  the  same  way  we  may  prove  that 


A  few  examples  of  this  sort  will  establish  what  is 
called  the  rule  of  signs  in  multiplication;  viz.,  that  a 
term  of  the  multiplicand  multiplied  by  a  term  of  the 
multiplier  has  the  sign  -j-  before  it  if  the  terms  have 
the  same  sign,  and  —  if  they  have  different  signs. 
But  here  the  student  must  avoid  using  an  incorrect 
mode  of  expression,  which  is  very  common,  viz.,  the 
saying  that  -f-  multiplied  by  -j-  gives  -)-;  —  multiplied 
by  -f-  gives  —  ;  and  so  on.  He  must  recollect  that 
the  signs  -(-  and  —  are  not  quantities,  but  directions 
to  add  and  subtract,  and  that,  as  has  been  well  said 
by  one  of  the  most  luminous  writers  on  algebra  in  our 
language,  we  might  as  well  say,  that  take  away  multi- 
plied by  take  away  gives  add,  as  that  —  multiplied  by 
—  gives  +.-* 

The  only  way  in  which  the  student  should  accus- 
tom himself  to  state  this  rule  is  the  following  :  "In 


*Frend,  Principles  of  Algebra.  The  author  of  this  treatise  is  far  from 
agreeing  with  the  work  which  he  has  quoted  in  the  rejection  of  the  isolated 
negative  sign  which  prevails  throughout  it,  but  fully  concurs  in  what  is  there 
said  of  the  methods  then  in  use  for  explaining  the  difficulties  of  the  negative 
sign. 


72  ON  THE  STUDY  OF  MATHEMATICS. 

multiplying  two  algebraical  expressions,  multiply  each 
term  of  the  one  by  each  term  of  the  other,  and  wher- 
ever two  terms  are  preceded  by  the  same  sign  put  -|- 
before  the  product  of  the  two;  when  the  signs  are 
different  put  the  sign  —  before  their  product.  " 

If  the  student  should  meet  with  an  equation  in 
which  positive  and  negative  signs  stand  by  them- 
selves, such  as 


let  him,  for  the  present,  reject  the  example  in  which 
it  occurs,  and  defer  the  consideration  of  such  equa- 
tions until  he  has  read  the  explanation  of  them  to 
which  we  shall  soon  come.  Above  all,  he  must  reject 
the  definition  still  sometimes  given  of  the  quantity 
—  a,  that  it  is  less  than  nothing.  It  is  astonishing  that 
the  human  intellect  should  ever  have  tolerated  such 
an  absurdity  as  the  idea  of  a  quantity  less  than  noth- 
ing ;  *  above  all,  that  the  notion  should  have  outlived 
the  belief  in  judicial  astrology  and  the  existence  of 
witches,  either  of  which  is  ten  thousand  times  more 
possible. 

These  remarks  do  not  apply  to  such  an  expression 
as  —  b  +  a,  which  we  sometimes  write  instead  of  a  — 
as  long  as  it  is  recollected  that  the  one  is  merely  used 
to  stand  for  the  other,  and  for  the  present  a  must  be 
considered  as  greater  than  b. 


•For  a  fall  critical  and  historical  diseass* 
Da  mJOults  Jmns  la  sdtmas  Je  rminmmttmmt 

edition,  Paris,  Gauthier-Villars,  1896).— £*»*r. 


ELEMENTARY  RULES  OF  ALGEBRA.  73 

In  writing  algebraical  expressions,  we  have  seen 
that  various  arrangements  may  be  adopted.  Thus 
ax2 — bx-\-  c  may  be  written  as  c-\-  ax*  —  bx,  or  — bx 
-\-c-\-ax2.  Of  these  three  the  first  is  generally  chosen, 
because  the  highest  power  of  x  is  written  first ;  the 
highest  but  one  comes  next ;  and  last  of  all  the  term 
which  contains  no  power  of  x.  When  written  in  this 
way  the  expression  is  said  to  be  arranged  in  descend- 
ing powers  of  x;  had  it  been  written  thus,  c — bx  -}-  ax2, 
it  would  have  been  arranged  in  ascending  powers  of 
x;  in  either  case  it  is  said  to  be  arranged  in  powers 
of  x,  which  is  called  the  principal  letter.  It  is  usual 
to  arrange  all  expressions  which  occur  in  the  same 
question  in  powers  of  the  same  letter,  and  practice 
must  dictate  the  most  convenient  arrangement.  Time 
and  trouble  is  saved  by  this  operation,  as  will  be  evi- 
dent from  multiplying  two  unarranged  expressions  to- 
gether, and  afterwards  doing  the  same  with  the  same 
expressions  properly  arranged. 

In  multiplying  two  arranged  expressions  together, 
while  collecting  such  terms  into  one  as  will  admit  of 
it,  it  will  always  be  evident  that  the  first  and  last  of 
all  the  products  contain  powers  of  the  principal  letter 
which  are  found  in  no  other  part,  and  stand  in  the 
product  unaltered  by  combination  with  any  other 
terms,  while  in  the  intermediate  products  there  are 
often  two  or  more  which  contain  the  same  power  of 
the  principal  letter,  and  can  be  reduced  into  one. 
This  will  be  evident  in  the  following  examples  : 


74 


ON  THE  STUDY  OF  MATHEMATICS. 


w 


w 

I 


4- 


X 

to 

4- 


4- 


CO 


4- 


H 

CD 


W 


bd  cr. 
*<* 


*. 

+  + 


fcO     CO 


00 


4 


I 

4 
+ 

/— • v 

4 
4- 

| 

+ 

<* 


4- 


It  is  plain  from  the  rule  of  multiplication,  that  the 
highest  power  of  x  in  a  product  must  be  formed  by 
multiplying  the  highest  power  in  one  factor  by  the 
highest  power  in  the  other,  or  when  the  two  factors 
have  been  arranged  in  descending  powers,  the  yfrj/ 
power  in  one  by  the  first  power  in  the  other.  Also, 
that  the  lowest  power  of  x,  or  should  it  so  happen, 


ELEMENTARY  RULES  OF  ALGEBRA.  75 

the  term  in  which  there  is  no  power  of  xt  is  made  by 
multiplying  the  last  terms  in  each  factor.  These  be- 
ing the  highest  and  lowest,  there  can  be  no  other  such 
power,  consequently  neither  of  these  terms  can  co- 
alesce with  any  other,  as  is  the  case  in  the  intermedi- 
ate products.  This  remark  will  be  of  most  convenient 
application  in  division,  to  which  we  now  come. 

Division  is  in  all  respects  the  reverse  of  multipli- 
cation. In  dividing  a  by  b  we  find  the  answer  to  this 
question  :  If  a  be  divided  into  b  equal  parts,  what  is 
the  magnitude  of  each  of  those  parts?  The  quotient 
is,  from  the  definition  of  a  fraction,  the  same  as  the 

fraction  —  ,  and  all  that  remains  is  to  see  whether  that 
b 

fraction  can  be  represented  by  a  simple  algebraical 
expression  without  fractions  or  not  ;  just  as  in  arith- 
metic the  division  of  200  by  26  is  the  reduction  of  the 
fraction  -2^  to  a  whole  number,  if  possible.  But  we 
must  here  observe  that  a  distinction  must  be  drawn 
between  algebraical  and  arithmetical  fractions.  For 


example,  —  ^  is  an  algebraical  fraction,  that  is,  there 
is  no  expression  without  fractions  which  is  always 

equal  to  —^—.    But  it  does  not  follow  from  this  that 
a  —  b  I    7 

the  number  which          T  represents  is  always  an  arith- 
a  —  b 

metical  fraction  ;  the  contrary  may  be  shown.     Let  a 

stand  for  12,  and  b  for  6,  then  --  -  is  3.     Again, 

a  —  b 

a?-\-ab  is  a  quantity  which  does  not  contain  algebrai- 
cal fractions,  but  it  by  no  means  follows  that  it  may 
not  represent  an  arithmetical  fraction.  To  show  that 


76 


ON  THE  STUDY  OF  MATHEMATICS. 


it  may,  let  a  =  %  and  l>  =  2,  then  a*-\-ab  =  \\  or  J. 
Other  examples  will  clear  up  this  point  if  any  doubt 
yet  exist  in  the  mind  of  the  student.  Nevertheless, 
the  following  propositions  of  arithmetic  and  algebra, 
which  only  differ  in  this,  that  "whole  number"  in  the 
arithmetical  proposition  is  replaced  by  "simple  ex- 
pression"* in  the  algebraical  one,  connect  the  two 
subjects  and  render  those  demonstrations  which  are 
in  arithmetic  confined  to  whole  numbers,  equally  true 
in  algebra  as  far  as  regards  simple  expressions  : 


The  sum,  difference,  or  pro- 
duct of  two  whole  numbers,  is  a 
whole  number. 

One  number  is  said  to  be  a 
measure  of  another  when  the 
quotient  of  the  two  is  a  whole 
number. 

The  greatest  common  meas- 
ure of  two  whole  numbers  is  the 
greatest  whole  number  which 
measures  both,  and  is  the  pro- 
duct of  all  the  prime  numbers 
which  will  measure  both. 

When  one  number  measures 
two  others,  it  measures  their 
sum,  difference,  and  product. 

In  the  division  of  one  number 
by  another,  the  remainder  is 
measured  by  any  number  which 
measures  the  dividend  and  di- 
visor. 


The  sum,  difference,  or  pro- 
duct of  two  simple  expressions 
is  a  simple  expression. 

One  expression  is  said  to  be  a 
measure  of  another  when  the 
quotient  of  the  two  is  a  simple 
expression. 

The  greatest  common  meas- 
ure of  two  expressions  is  the 
common  measure  which  has  the 
highest  exponents  and  coeffi- 
cients, and  is  the  product  of  all 
prime  simple  expressions  which 
measure  both. 

When  one  expression  meas- 
ures two  others,  it  measures 
their  sum,  difference,  and  pro- 
duct. 

In  the  division  of  one  expres- 
sion by  another,  the  remainder 
is  measured  by  any  expression 
which  measures  the  dividend 
and  divisor. 


*By  a  simple  expression  is  meant  one  which  does  not  contain  the  princi- 
pal letter  in  the  denominator  of  any  fraction. 


ELEMENTARY  RULES  OP  ALGEBRA.  77 

A  fraction  is  not  altered  by  A  fractional  expression  is  not 
multiplying  or  dividing  both  its  altered  by  multiplying  or  divid- 
numerator  and  denominator  by  ing  both  its  numerator  and  de- 
the  same  quantity.  nominator  by  the  same  expres- 

sion. 

In  the  term  simple  expression  are  included  those 
quantities  which  contain  arithmetical  fractions,  pro- 
vided there  is  no  algebraical  quantity,  or  quantity  rep- 
resented by  letters  in  the  denominator;  thus  \ab-\-\ 
is  called  a  simple  expression.  We  now  proceed  to 
the  division  of  one  simple  expression  by  another,  and 
we  will  take  first  the  case  where  neither  quantity  con- 
tains more  than  one  term.  For  example,  what  is 
42  04  ft  c  divided  by  6  a2  b  cl  that  is,  what  quantity  must 
be  multiplied  by  §cPbcy  in  order  to  produce  4204£3<r. 
This  last  expression  written  at  length,  \s4£aaaabbbc, 
and  42  is  6x7.  We  can  then  separate  this  expression 
into  the  product  of  two  others,  one  of  which  shall  be 
6  a2  b  c,  or  6  a  a  b  c ;  it  will  then  be  Qaabcy^l aabb, 
and  it  is  7 aabb  which  must  be  multiplied  by  Saabc 
in  order  to  produce  42  #4  b*  c.  A  few  examples  worked 
in  this  way,  will  lead  the  student  to  the  rule  usually 
given  in  all  cases  but  one,  to  which  we  now  come. 
We  have  represented  cc,  ccc,  cccc,  etc.,  by  c2,  <?,  c*, 
etc.,  and  have  called  them  the  second,  third,  fourth, 
etc.,  powers  of  c.  The  extension  of  this  rule  would 
lead  us  to  represent  c  by  c1,  and  call  it  the  first  power 
of  c.  Again,  we  have  represented  c  -j-  c,  c  -\-  c  -\-  c, 
c  -j-  c  -(-  c  -|-  c,  etc.  by  2^-,  3<r,  4^-,  and  have  called  2,  3, 
4,  etc.,  the  coefficients  of  c.  The  extension  of  this 


78  ON  THE  STUDY  OF  MATHEMATICS. 

rule  would  lead  us  to  write  c  thus,  lc,  or,  rather,  if  we 
attend  to  the  last  remark,  l^1.  This  instance  leads  us 
to  observe  the  gradual  progress  of  our  language.  We 
begin  with  the  quantity  c  by  itself ;  we  proceed  in  our 
course,  shortening  by  new  signs  the  more  complicated 
combinations  of  c,  and  the  original  quantity  c  forces 
itself  anew  upon  our  attention  as  a  part  of  the  series, 

c,  2  <r,  3  c,  4  <r,  etc. ,  and  <r,  <r2,  <r3,  <r4,  etc. , 
in  each  of  which,  except  the  first,  there  is  a  distinct 
figure,  which  is  called  a  coefficient  or  exponent,  ac- 
cording to  its  situation.  We  then  deduce  rules  in 
which  the  terms  coefficient  or  exponent  occur,  but 
which,  of  course,  cannot  apply  to  the  first  term  in 
each  series,  because,  as  yet,  it  has  neither  coefficient 
nor  exponent.  Among  such  rules  are  the  following  : 
I.  To  add  two  terms  of  the  first  series,  add  the  co- 
efficients, and  affix  to  the  sum  the  letter  c.  Thus 
±c-\-Z<;  =  fl  c.  II.  To  multiply  two  terms  of  the  sec- 
ond series,  add  the  exponents,  and  make  this  sum  the 
exponent  of  c.  Thus  ^X^  =  ^-  HI.  To  divide  a  term 
of  the  second  series  by  one  which  comes  before  it,  sub- 
tract the  exponent  of  the  divisor  from  the  exponent 
of  the  dividend,  and  make  this  difference  the  exponent 
of  c.  Thus,  c1 

-?=*• 

These  rules  are  intelligible  for  all  terms  of  the 
series  except  the  first,  to  which,  nevertheless,  they 
will  apply  if  we  agree  that  lcl  shall  represent  c,  as 
will  be  evident  by  applying  either  of  them  to  find 


ELEMENTARY  RULES  OF  ALGEBRA.  79 

f\ 

j  or  —.  We  therefore  agree  that  I*:1  shall 
stand  for  c,  and  although  c  is  not  written  thus,  it  must 
be  remembered  that  c  is  to  be  considered  as  having 
the  coefficient  1  and  the  exponent  1,  which  is  an 
amendment  and  enlargement  of  our  algebraical  lan- 
guage, derived  from  experience.  It  may  be  said  that 
this  is  all  superfluous,  because,  if  c^  stand  for  cc,  and 
fl  for  ccc,  what  can  c1  stand  for  but  <r?  But  it  must 
be  recollected  that,  since  the  symbol  c1  has  not  yet  re- 
ceived a  meaning,  we  are  at  liberty  to  make  it  stand 
for  anything  which  we  please,  for  example,  for—  -  , 
or  c — r2,  or  any  other.  If  we  did  this,  there  would, 
indeed,  be  a  great  violation  of  analogy,  that  is,  what 
c1  stands  for  would  not  be  as  like  that  which  c2  has 
been  made  to  stand  for,  as  the  meaning  of  c*  is  to 
that  of  c^j  but,  nevertheless,  we  should  not  be  led  to 
any  incorrect  results  as  long  as  we  remembered  to 
make  c1  always  stand  for  the  same  thing.  These  re- 
marks are  here  introduced  in  order  to  show  the  man- 
ner in  which  analogy  is  followed  in  extending  the  lan- 
guage of  algebra,  and  to  prove  that,  after  a  certain 
period,  we  may  rather  be  said  to  discover  new  symbols 
than  to  make  them.  The  immense  importance  of  this 
branch  of  the  subject  makes  it  necessary  that  it  should 
be  fully  and  early  understood  by  all  who  intend  to 
pursue  their  mathematical  studies  to  any  depth.  To 
illustrate  it  still  further,  we  subjoin  another  instance, 
which  has  not  been  noticed  in  its  proper  place. 


8o  ON  THE  STUDY  OF  MATHLMATICS, 

The  signs  -f  and  —  were  first  used  to  connect  one 
quantity  with  others,  and  to  show  what  arithmetical 
operations  were  performed  on  other  quantities  by 
means  of  the  first.  But  the  first  quantity  on  which 
we  begin  the  operation  is  not  preceded  by  any  sign, 
not  being  considered  as  added  to  or  subtracted  from 
any  previous  one.  Rules  were  afterwards  deduced  for 
the  addition  and  subtraction  of  the  total  result  of  sev- 
eral expressions  in  which  these  signs  occur,  as  follows : 

To  add  two  expressions,  form  a  third,  which  has 
all  the  quantities  in  the  first  two,  with  the  same  signs. 

To  subtract  one  expression  from  another,  change 
the  sign  of  each  term  of  the  subtrahend,  and  proceed 
as  in  the  last  rule. 

The  only  terms  in  which  these  rules  do  not  apply 
are  those  which  have  no  sign,  viz.,  the  first  of  each. 
But  they  will  apply  to  those  terms,  and  will  produce 
correct  results,  if  we  place  the  sign  -f  before  each  of 
them.  We  are  thus  led  to  see  that  an  algebraical 
term  which  has  no  sign  is  equivalent  in  all  operations 
to  one  which  is  preceded  by  the  sign  -f-.  We,  there- 
fore, consider  this  sign  as  prefixed,  though  it  is  not 
always  written,  and  thus  we  are  furnished  with  a 
method  of  containing  under  one  rule  that  which  would 
otherwise  require  two. 

From  these  considerations  the  following  appears 
to  be  the  best  and  most  natural  course  of  proceeding 
in  the  invention  of  additional  symbols.  When  a  rule 
has  been  discovered  which  is  not  quite  general,  and 


ELEMENTARY  RULES  OF  ALGEBRA.  81 

which  only  fails  in  its  application  to  a  few  instances, 
annex  such  additional  symbols  to  those  already  in  use, 
or  change  and  modify  these  so  as  to  make  the  rule 
applicable  in  all  cases,  provided  always  this  can  be 
done  without  making  the  same  symbol  stand  for  two 
different  things,  and  without  any  violation  of  analogy. 
If  the  rule  itself,  by  its  application  to  any  case,  should 
produce  a  new  symbol  hitherto  unexplained,  it  is  a 
sign  that  the  rule  has  been  applied  to  a  case  which 
was  never  intended  to  fall  under  it  when  it  was  made. 
For  the  solution  of  this  case  we  must  have  recourse 
to  first  principles,  but  when,  by  these  means,  the  re- 
sult has  been  found,  it  will  be  best  to  agree  that  the 
new  symbol  furnished  by  the  rule  shall  stand  for  the 
result  furnished  by  the  principle,  by  which  means  the 
generality  of  the  rule  will  be  attained  and  the  analogy 
of  language  will  not  be  injured.  Of  this  the  following 
is  a  remarkable  instance  : 

To  divide  c8  by  <r5  the  rule  tells  us  to  subtract  5 
from  8,  and  make  the  result  the  exponent  of  c,  which 
gives  the  quotient  <?.  If  we  apply  the  same  rule  to  di- 
vide c*  by  <r6,  since  6  subtracted  from  6  leaves  0,  the 
result  is  <r°,  a  new  symbol,  to  which  we  have  attached 
no  meaning.  The  fact  is  that  the  rule  was  formed 
from  observation  of  different  powers  of  <r,  and  was 
never  intended  to  apply  to  the  division  of  a  power  of 
c  by  the  same  power.  If  we  apply  the  common  prin- 
ciples to  the  division  of  <:6  by  c6,  the  result  is  1.  We, 
therefore,  agree  that  <r°  shall  stand  for  1,  and  the  least 


82  ON  THE  STUDY  OF  MATHEMATICS. 

inspection  will  show  that  this  agreement  does  not  af- 
fect the  truth  of  any  result  derived  from  the  rule.  If, 
in  the  solution  of  any  problem,  the  symbol  <r°  should 
appear,  we  must  consider  it  is  a  sign  that  we  have,  in 
the  course  of  the  investigation,  divided  a  power  of  c 
by  itself  by  the  common  rule,  without  remarking  that 
the  quotient  is  1.  We  must,  therefore,  replace  <r°  by 
1-,  but  it  is  entirely  indifferent  at  what  stage  of  the 
process  this  is  done. 

Several  extensions  might  be  noticed,  which  are 
made  almost  intuitively,  to  which  these  observations 
will  apply.  Such,  for  example,  is  the  multiplication 
and  division  of  any  number  by  1,  which  is  not  con- 
templated in  the  definition  of  these  operations.  Such 
is  also  the  continual  use  of  0  as  a  quantity,  the  addi- 
tion and  subtraction  of  it  from  other  quantities,  and 
the  multiplication  of  it  by  others,  neither  of  which 
were  contemplated  when  these  operations  were  first 
thought  of. 

We  now  proceed  to  the  principles  on  which  more 
complicated  divisions  are  performed.  The  question 
proposed  in  division,  and  the  manner  of  answering  it, 
may  be  explained  in  the  following  manner.  Let  A  be 
an  expression  which  is  to  be  divided  by  If,  and  let  Q 
be  the  quotient  of  the  two.  By  the  meaning  of  divi- 
sion, if  there  be  no  remainder  A  =  Qff,  since  the  quo- 
tient is  the  expression  which  must  multiply  the  di- 
visor, in  order  to  produce  the  dividend.  Now  let  the 


ELEMENTARY  RULES  OF  ALGEBRA.  83 

quotient  be  made  up  of  different  terms,  a,  b,  c,  etc., 
let  it  be  a  +  b  —  c  +  d.  That  is,  let 

A  =  QH  (1) 

Q  =  a  +  b  —  c  +  d.  (2) 

By  putting,  instead  of  Q  in  (1),  that  which  is  equal 
to  it  in  (2),  we  find 

A  =  (a+b  —  c  +  d}H=aH+bH—cH+dH  (3) 
Now  suppose  that  we  can  by  any  method  find  the 
term  a  of  the  quotient,  that  is,  that  we  can  by  trial  or 
otherwise  find  one  term  of  the  quotient.  In  (3),  when 
the  term  a  is  found,  since  H  is  known,  the  term  aH 
is  found.  Now  if  two  quantities  are  equal,  and  from 
them  we  subtract  the  same  quantity,  the  remainders 
will  be  equal.  Subtract  affirom  the  equal  quantities 
A  and  aH+  bH—cH+  dH,  and  we  shall  find 

A  —  aH=  bH—  cH+  dH=  (b  —  c  +  d}H.  (4) 
If,  then,  we  multiply  the  term  of  the  quotient  found 
by  the  divisor,  and  subtract  the  product  from  the  divi- 
dend, and  call  the  remainder  B;  then 

B=(b  —  c  +  d}H.  (5) 

That  is,  if  B  be  made  a  dividend,  and  H  still  continue 
the  divisor,  the  quotient  is  b  —  c  -j-  d,  or  all  the  first 
quotient,  except  the  part  of  it  which  we  have  found. 
We  then  proceed  in  the  same  manner  with  this  new 
dividend,  that  is,  we  find  b  and  also  bH,  and  subtract 
it  from  B,  and  let  B — bHbe  represented  by  C,  which 
gives  by  the  process  which  has  just  been  explained 

C=(  —  c  +  d}H=—cH+dH.  (6) 

We  now  come  to  a  negative  term  of  the  quotient. 


84  ON  THE  STUDY  OF  MATHEMATICS. 

Let  us  suppose  that  we  have  found  c,  and  that  its  sign 
in  the  quotient  is  —  .  If  two  quantities  are  equal,  and 
we  add  the  same  quantity  to  both,  the  sums  are  equal. 
Let  us  therefore  add  cH  to  both  the  equal  quantities 
in  (6),  and  the  equation  will  become 

C+cH=dH;  (7) 

or  if  we  denote  C-\-  cH  by  Z>,  this  is 


There  is  only  one  term  of  the  quotient  remaining,  and 
if  that  can  be  found  the  process  is  finished.  But  as 
we  cannot  know  when  we  have  come  to  the  last  term, 
we  must  continue  the  same  process,  that  is,  subtract 
dH  from  D,  in  doing  which  we  shall  find  that  dH  is 
equal  to  D,  or  that  the  remainder  is  nothing.  This 
indicates  that  the  quotient  is  now  exhausted  and  that 
the  process  is  finished. 

We  will  now  apply  this  to  an  example  in  which 
the  quotient  is  of  the  same  form  as  that  in  the  last 
process,  namely,  consisting  of  four  terms,  the  third  of 
which  has  the  negative  sign.  This  is  the  division  of 

#4  —  /  —  3  x^y2  +  x*y+2  xy*  by  x  —y. 
Arrange  the  first  quantity  in  descending  powers  of  x 
which  will  make  it  stand  thus  : 

x*  _|_  y*y  —  3  *2/  -j-  2  xy*  —y*  (A) 

One  term  of  the  quotient  can  be  found  immediately, 
for  since  it  has  been  shown  that  the  term  containing 
the  highest  power  of  x  in  a  product  is  made  up  of 
nothing  but  the  product  of  the  terms  containing  the 
highest  powers  of  x  which  occur  in  the  multiplier  and 


ELEMENTARY  RULES  OF  ALGEBRA. 


multiplicand,  and  considering  that  the  expression  (A) 
is  the  product  of  x — y  and  the  quotient,  we  shall  re- 
cover the  highest  power  of  x  in  the  quotient  by  divid- 
ing x*,  the  highest  power  of  x  in  (A),  by  x,  its  highest 
power  in  x — y.  This  division  gives  x3  as  the  first 
term  of  the  quotient.  The  following  is  the  common 
process,  and  with  each  line  is  put  the  corresponding 
step  of  the  process  above  explained,  of  which  this  is 
an  example : 


0) 

P 

w' 


i- 


hrj  H  C/5 

1  £  § 

P-  p 

IT.  C  C/)  D,  V)  ^        C/) 

C     Cu  C  ""i*  £  p.          pi 

cr  £•  cr  2.  cr  5- 

£  2.  a;  a  ft  3.      fT 

rj       O-1  Q3  CD  £3  fK  pj 

o    2  o  P  Q.  n>        o 


!  fc 


++ 


to  to 

H»  Ho 


to  cc 

L>        L> 

to      w 


to 


H. 


Vi 


CO 


s 


to 

5"^ 

Ve  *~s 


86  ON  THE  STUDY  OF  MATHEMATICS. 

The  second  and  following  terms  of  the  quotient 
are  determined  in  exactly  the  same  manner  as  the 
first.  In  fact,  this  process  is  not  the  finding  of  a  quo- 
tient directly  from  the  divisor  and  dividend,  but  one 
term  is  first  found,  and  by  means  -of  that  term  another 
dividend  is  obtained,  which  only  differs  from  the  first 
in  having  one  term  less  in  the  quotient,  viz.,  that 
which  was  first  found.  From  this  second  dividend 
one  term  of  its  quotient  is  found,  and  so  on  until  we 
obtain  a  dividend  whose  quotient  has  only  one  term, 
the  finding  of  which  finishes  the  process.  It  is  usual 
also  to  neglect  all  the  terms  of  the  first  dividend, 
except  those  which  are  immediately  wanted,  taking 
down  the  others  one  by  one  as  they  become  necessary. 
This  is  a  very  good  method  in  practice  but  should  be 
avoided  in  explaining  the  principle,  since  the  first 
subtraction  is  made  from  the  whole  dividend,  though 
the  operation  may  only  affect  the  form  of  some  part 
of  it. 

If  the  student  will  now  read  attentively  what  has 
been  said  on  the  greatest  common  measure  of  two 
numbers,  and  then  examine  the  connexion  of  whole 
numbers  in  arithmetic  and  simple  expressions  in  alge- 
bra with  which  we  commenced  the  subject  of  division, 
he  will  see  that  the  greatest  algebraical  common  meas- 
ure of  two  expressions  may  be  found  in  exactly  the 
same  manner  as  the  same  operation  is  performed  in 
arithmetic.  He  must  also  recollect  that  the  greatest 
common  measure  of  two  expressions  A  and  B  is  not 


ELEMENTARY  RULES  OF  ALGEBRA.          87 

altered  by  multiplying  or  dividing  either  of  them,  A, 
for  example,  by  any  quantity,  provided  that  quantity 
has  no  measure  in  common  with  B.  For  example, 
the  greatest  common  measure  of  a2  —  x2  and  bcfi  —  bx* 
is  the  same  with  that  of  2  a?  —  2x2  and  aB — x3,  since 
though  a  new  measure  is  now  introduced  into  the  first 
and  taken  away  from  the  second,  nothing  is  introduced 
or  taken  away  which  is  common  to  both.  The  same 
observation  applies  to  arithmetic  also.  For  example, 
take  the  numbers  162  and  180.  We  may,  without 
altering  their  greatest  common  measure,  multiply  the 
first  by  7  and  the  second  by  11,  etc.  The  rule  for 
finding  the  greatest  common  measure  should  be  prac- 
tised with  great  attention  by  all  who  intend  to  proceed 
beyond  the  usual  stage  in  algebra.  To  others  it  is  not 
of  the  same  importance,  as  the  necessity  for  it  never 
occurs  in  the  lower  branches  of  the  science. 

In  proceeding  to  the  subject  of  fractions,  it  must 
be  observed  that,  in  the  same  manner  as  in  arithmetic, 
when  there  is  a  remainder  which  cannot  be  further 
divided  by  the  divisor,  that  is,  where  the  dividend  is 
so  reduced  that  no  simple  term  multiplied  by  the  first 
term  of  the  divisor  will  give  the  first  term  of  the  re- 
mainder, as  in  the  case  where  the  divisor  is  a2  x  -\-bx2 
and  the  remainder  ax-\-  b;  in  this  case  a  fraction 
must  be  added  to  the  quotient,  whose  numerator  is 
this  remainder,  and  whose  denominator  is  the  divisor. 
Thus,  in  dividing  a*-\-fi*  by  a-\-b,  the  quotient  is 
a3  —  a?b-\-ab2  —  b*,  and  the  remainder  2  £4,  whence 


88  ON  THE  STUDY  OF  MATHEMATICS. 


The  arithmetical  rules  for  the  addition,  etc.,  of  frac- 
tions hold  equally  good  when  the  numerators  and  de- 

3  1 

nominators  are  themselves  fractions.     Thus  -|  and  -| 

T  ^ 

are  added,  etc.,  exactly  in  the  same  way  as  |  and  fy, 
the  sum  of  the  second  being 

7X2+5X3 

5X7 
and  that  of  the  first 

|X|+fX» 

»x* 

The  rules  for  the  addition,  etc.,  of  algebraic  fractions 
are  exactly  the  same  as  in  arithmetic  ;  for  both  the 
numerator  and  denominator  of  every  algebraic  frac- 
tion stands  either  for  a  whole  number  or  a  fraction, 
and  therefore  the  fraction  itself  is  either  of  the  same 

2 

form  as  f  or-|.  Nevertheless  the  student  should  at- 
tend to  some  examples  of  each  operation  upon  alge- 
braic fractions,  by  way  of  practice  in  the  previous 
operations.  As  the  subject  is  not  one  which  presents 
any  peculiar  difficulties,  we  shall  now  pass  on  to  the 
subject  of  equations,  concluding  this  article  with  a 
list  of  formulae  which  it  is  highly  desirable  that  the 
student  should  commit  to  memory  before  proceeding 
to  any  other  part  of  the  subject. 

(a +  *)  +  (*  —  *)  =  2*  (1) 

(a  +  t)  —  (a  —  f)  =  26  (2) 

a—(a  —  V)  =  b  (3) 


ELEMENTARY  RULES  OF  ALGEBRA. 


(2*  #  +  £)2=4tf2*2-f  ±abx  +  P 

(6) 

(a-\-b}(a  —  &)  =  <&  —  & 

(7) 

((:i:x(:_?=J-E:+?:+:'} 

(8) 

a        ma 
b        mb 

(9) 

!        <T               #</-{-<:                          ^               0d?  C 

(10) 

d                      d          '                     d                      d 

a    (     c         ad-\-bc       a         c         ad  —  be 

(11) 

b         d  ~          bd      '      b         d  ~          bd 

a               ac        a       a         c        ac 

(12) 

c 

a                   a 

b         a         c 

(13) 

a                     a 

b        ad        c 
c         be        ~b 

(14) 

~d                    ~d 

1     _  b 

a  ~~  a 

(15} 

CHAPTER  VIII. 

EQUATIONS  OF  THE  FIRST  DEGREE. 

WE  have  already  defined  an  equation,  and  have 
come  to  many  equations  of  different  sorts.  But 
all  of  them  had  this  character,  that  they  did  not  de- 
pend upon  the  particular  number  which  any  letter 
stood  for,  but  were  equally  true,  whatever  numbers 
might  be  put  in  place  of  the  letters.  For  example,  in 
the  equation 


the  truth  of  the  assertion  made  in  this  algebraical  sen- 
tence is  the  same,  whether  a  be  considered  as  repre- 
senting 1,  2,  2£,  etc.,  or  any  other  number  or  fraction 
whatever.  The  second  side  of  this  equation  is,  in 
fact,  the  result  of  the  operation  pointed  out  on  the 
first  side.  On  the  first  side  you  are  directed  to  divide 
a2  —  1  by  a-\-  1  ;  the  second  side  shows  you  the  result 
of  that  division.  An  equation  of  this  description  is 
called  an  identical  equation,  because,  in  fact,  its  two 
sides  are  but  different  ways  of  writing  down  the  same 


EQUATIONS  OF  THE  FIRST  DEGREE.  QI 

number.     This  will  be  more  clearly  seen  in  the  iden- 

tical equations 

0  +  0  =  20,  70  —  30  +  ^  =  40  — 


. 
o 

The  whole  of  the  formulae  at  the  end  of  the  last 
article  are  examples  of  identical  equations.  There  is 
not  one  of  them  which  is  not  true  for  all  values  which 
can  be  given  to  the  letters  which  enter  into  them,  pro- 
vided only  that  whatever  a  letter  stands  for  in  one 
part  of  an  equation,  it  stands  for  the  same  in  all  the 
other  parts. 

If  we  take,  now,  such  an  equation  as  0+1  =  8,  we 
have  an  equation  which  is  no  longer  true  for  every 
value  which  can  be  given  to  its  algebraic  quantities. 
It  is  evident  that  the  only  number  which  a  can  repre- 
sent consistently  with  this  equation  is  7,  as  any  other 
supposition  involves  absurdity.  This  is  a  new  spe- 
cies of  equation,  which  can  only  exist  in  some  partic- 
ular case,  which  particular  case  can  be  found  from 
the  equation  itself.  The  solution  of  every  problem 
leads  to  such  an  equation,  as  will  be  shown  hereafter, 
and,  in  the  elements  of  algebra,  this  latter  species  of 
equation  is  of  most  importance.  In  order  to  distin- 
guish them  from  identical  equations,  they  are  called 
equations  of  condition,  because  they  cannot  be  true  when 
the  letters  contained  in  them  stand  for  any  number 
whatever,  and  their  very  existence  makes  a  condition 
which  the  letters  contained  must  fulfil.  The  solution 
of  an  equation  of  condition  is  the  process  of  finding 


Q2  ON  THE  STUDY  OF  MATHEMATICS. 

what  number  the  letter  must  stand  for  in  order  that 
the  equation  may  be  true.  Every  such  solution  is  a 
process  of  reasoning,  which,  setting  out  with  suppos- 
ing the  truth  of  the  equation,  proceeds  by  self-evident 
steps,  making  use  of  the  common  rules  of  arithmetic 
and  algebra.  We  shall  return  to  the  subject  of  the 
solution  of  equations  of  condition,  after  showing,  in  a 
few  instances,  how  we  come  to  them  in  the  solution 
of  problems.  In  equations  of  condition,  the  quantity 
whose  value  is  determined  by  the  equation  is  usually 
represented  by  one  of  the  last  letters  of  the  alphabet, 
and  all  others  by  some  of  the  first.  This  distinction 
is  necessary  only  for  the  beginner ;  in  time  he  must 
learn  to  drop  it,  and  consider  any  letter  as  standing 
for  a  quantity  known  or  unknown,  according  to  the 
conditions  of  the  problem. 

In  reducing  problems  to  algebraical  equations  no 
general  rule  can  be  given.  The  problem  is  some  prop- 
erty of  a  number  expressed  in  words  by  which  that 
number  is  to  be  found,  and  this  property  must  be 
written  down  as  an  equation  in  the  most  convenient 
way.  As  examples  of  this,  the  reduction  of  the  fol- 
lowing problems  into  equations  is  given  : 

I.  What  number  is  that  to  which,  if  56  be  added, 
the  result  will  be  200  diminished  by  twice  that  num- 
ber? 

Let  x  stand  for  the  number  which  is  to  be  found. 

Then  *  +  56  =  200  —  2x. 

If,  instead  of  56,  200,  and  2,  any  other  given  num- 


EQUATIONS  OF  THE  FIRST  DEGREE.  93 

bers,  a,  b,  and  c,  are  made  use  of  in  the  same  man- 
ner, the  equation  which  determines  x  is 

x-\-  a  =  b —  ex. 

II.  Two  couriers  set  out  from  the  same  place,  the 
second  of  whom  goes  three  miles  an  hour,  and  the 
first  two.  The  first  has  been  gone  four  hours,  when 
the  second  is  sent  after  him.  How  long  will  it  be  be- 
fore he  overtakes  him  ? 

Let  x  be  the  number  of  hours  which  the  second 
must  travel  to  overtake  the  first.  At  the  time  when 
this  event  takes  place,  the  first  has  been  gone  x  -J-  4 
hours,  and  will  have  travelled  (^  +  4)2,  or  2x-\-S 
miles.  The  second  has  been  gone  x  hours,  and  will 
have  travelled  3  x  miles.  And,  when  the  second  over- 
takes the  first,  they  have  travelled  exactly  the  same 
distance,  and,  therefore, 

3x  =  2x  +  8. 

If,  instead  of  these  numbers,  the  first  goes  a  miles 
an  hour,  the  second  b,  and  c  hours  elapse  before  the 
second  is  sent  after  the  first, 

bx  =  ax  -f-  ac. 

Four  men,  A,  B,  C,  and  D,  built  a  ship  which 
cost  ;£2607,  of  which  B  paid  twice  as  much  as  A,  C 
paid  as  much  as  A  and  B,  and  D  as  much  as  C  and 
B.  What  did  each  pay? 

Suppose  that  A  paid  x  pounds, 
then  B  paid  2x  .  .  . 

C  paid  x  -f  2  x  or  3  x  .  .  . 
D  paid  2x-\-  3x  or  §x  .  .  . 


94  ON  THE  STUDY  OF  MATHEMATICS. 

All  together  paid  x  +  2x-\-3x+5x,  or  11  x,  there- 
fore 


There  are  two  cocks,  from  the  first  of  which  a  cis- 
tern is  filled  in  12  hours,  and  the  second  in  15.  How 
long  would  they  be  in  filling  it  if  both  were  opened 
together? 

Let  x  be  the  number  of  hours  which  would  elapse 
before  it  was  filled.  Then,  since  the  first  cock  fills 
the  cistern  in  12  hours,  in  one  hour  it  fills  T^  of  it,  in 
two  hours  T2^,  etc.,  and  in  x  hours  -f^.  Similarly,  in 
x  hours,  the  second  cock  fills  /F  of  the  cistern.  When 
the  two  have  exactly  filled  the  cistern,  the  sum  of 
these  fractions  must  represent  a  whole  or  1,  and, 
therefore, 

-  +  --1 
12  ^  15  ~ 

If  the  times  in  which  the  two  can  fill  the  cistern  are  a 
and  b  hours,  the  equation  becomes 

f+f-1- 

A  person  bought  8  yards  of  cloth  for  ^"3  2s.,  giving 
9s.  a  yard  for  some  of  it  and  7s.  a  yard  for  the  rest  ; 
how  much  of  each  sort  did  he  buy  ? 

Let  x  be  the  number  of  yards  at  7s.  Then  7  x  is 
the  number  of  shillings  they  cost.  Also  8  —  x  is  the 
number  of  yards  at  9s.,  and  (8  —  #)g,  or  72  —  9*,  is 
the  number  of  shillings  they  cost.  And  the  sum  of 


EQUATIONS  OF  THE  FIRST  DEGREE.  95 

these,  or  7jc-j-72  —  9#,  is  the  whole  price,  which  is 
;£3  2s.,  or  62  shillings,  and,  therefore, 

7*4-72  — 9*=:62. 

These  examples  will  be  sufficient  to  show  the 
method  of  reducing  a  problem  to  an  equation.  As- 
suming a  letter  to  stand  for  the  unknown  quantity,  by 
means  of  this  letter  the  same  quantity  must  be  found 
in  two  different  forms,  and  these  must  be  connected 
by  the  sign  of  equality.  However,  the  reduction  into 
equations  of  such  problems  as  are  usually  given  in  the 
treatises  on  algebra  rarely  occurs  in  the  applications 
of  mathematics.  The  process  is  a  useful  exercise  of 
ingenuity,  but  no  student  need  give  a  great  deal  of 
time  to  it.  Above  all,  let  no  one  suppose,  because  he 
finds  himself  unable  to  reduce  to  equations  the  conun- 
drums with  which  such  books  are  usually  filled,  that, 
therefore,  he  is  not  made  for  the  study  of  mathemat- 
ics, and  should  give  it  up.  His  future  progress  de- 
pends in  no  degree  upon  the  facility  with  which  he 
discovers  the  equations  of  problems  ;  we  mean  as  far 
as  power  of  comprehending  the  subsequent  sciences 
is  concerned.  He  may  never,  perhaps,  make  any  con- 
siderable step  for  himself,  but,  without  doing  this,  he 
may  derive  all  the  benefits  which  the  study  of  mathe- 
matics can  afford,  and  even  apply  them  extensively. 
There  is  nothing  which  discourages  beginners  more 
than  the  difficulty  of  reducing  problems  to  equations, 
and  yet,  as  respects  its  utility,  if  there  be  anything 
in  the  elements  which  may  be  dispensed  with,  it  is 


96  ON  THE  STUDY  OF  MATHEMATICS. 

this.  We  do  not  wish  to  depreciate  its  utility  as  an 
exercise  for  the  mind,  or  to  hinder  all  from  attempt- 
ing to  conquer  the  difficulties  which  present  them- 
selves ;  but  to  remind  every  one  that,  if  he  can  read 
and  understand  all  that  is  set  before  him,  the  essen- 
tial benefit  derived  from  mathematical  studies  will  be 
gained,  even  though  he  should  never  make  one  step 
for  himself  in  the  solution  of  any  problem. 

We  return  now  to  the  solution  of  equations  of  con- 
dition. Of  these  there  are  various  classes.  Equations 
of  the  first  degree,  commonly  called  simple  equations, 
are  those  which  contain  only  the  first  power  of  the  un- 
known quantity.  Of  this  class  are  all  the  equations 
to  which  we  have  hitherto  come  in  the  solution  of 
problems.  The  principle  by  which  they  are  solved  is, 
that  two  equal  quantities  may  be  increased  or  dimin- 
ished, multiplied,  or  divided  by  any  quantity,  and  the 
results  will  be  the  same.  In  algebraical  language, 
if  a  =  bt  a-\-  c  =  b-\-c,  a  —  <r  —  £  —  c,  ac  =  bc,  and 


-  =  —  .  In  every  elementary  book  it  is  stated  that 
any  quantity  may  be  removed  from  one  side  of  the 
equation  to  the  other,  provided  its  sign  be  changed. 
This  is  nothing  but  an  application  of  the  principle 
just  stated,  as  may  be  shown  thus  :  Let  a-\-  b  —  c  =  dv 
add  c  to  both  quantities,  then 

a-\-b  —  <r-f  c  =  d-}-c  or  a-\-b  = 


Again  subtract  b  from  both  quantities,  then  a-\-b  — 
c  —  b  =  d  —  b,   or  a  —  c  =  d  —  b.     Without  always  re- 


EQUATIONS  OF  THE  FIRST  DEGREE.  97 

peating  the  principle,  it  is  derived  from  observation, 
that  its  effect  is  to  remove  quantities  from  one  side  of 
an  equation  to  another,  changing  their  sign  at  the 
same  time.  But  the  beginner  should  not  use  this  rule 
until  he  is  perfectly  familiar  with  the  manner  of  using 
the  principle.  He  should,  until  he  has  mastered  a 
good  many  examples,  continue  the  operation  at  full 
length,  instead  of  using  the  rule,  which  is  an  abridg- 
ment of  it.  In  fact  it  would  be  better,  and  not  more 
prolix,  to  abandon  the  received  phraseology,  and  in 
the  example  just  cited,  instead  of  saying  "  bring  the 
term  b  to  the  other  side  of  the  equation,"  to  say  " sub- 
tract b  from  both  sides,"  and  instead  of  saying  "bring 
c  to  the  other  side  of  the  equation,"  to  say  "  add  c  to 
both  sides." 

Suppose  we  have  the  fractions  f ,  ^,  and  T5¥.  If  we 
multiply  them  all  by  the  product  of  the  denominators 
4x^X14,  or  392,  all  the  products  will  be  whole  num- 

TU           -11    u     3X392    1X392         ,5x392 
bers.     They   will    be  — ~ ,  — — = ,  and  — — —  , 

and  since  392  is  measured  by  4,  3  X  392  is  also  meas- 

3  V  392 
ured  by  4,  and  — ~-  is  a  whole  number,  and  so  on. 

But  any  common  multiple  of  4,  7,  and  14  will  serve 
as  well.  The  least  common  multiple  will  therefore  be 
the  most  convenient  to  use  for  this  purpose.  The 
least  common  multiple  of  4,  7,  and  14  is  28,  and  if  the 
three  fractions  be  multiplied  by  28,  the  results  will  be 
whole  numbers.  The  same  also  applies  to  algebraic 

fractions.     Thus  - ,  —  ,  and  -7-7^,  will  become  simple 
b    ae  bdj 


98  ON  THE  STUDY  OF  MATHEMATICS. 

expressions,  if  they  are  multiplied  by  b  X  <^X  bdfy  or 
ft  d**  ef.  But  the  most  simple  common  multiple  of  b, 
de,  and  bdf,  is  bdef,  which  should  be  used  in  pref- 
erence to  t>2  d*  ef. 

This  being  premised,  we  can  now  reduce  any  equa- 
tion which  contains  fractions  to  one  which  does  not. 
For  example,  take  the  equation 

x    ,    2*_     7         3  —  2* 

"3""F:~To~    ~~6 ' 

If  we  multiply  both  these  equal  quantities  by  any 
other,  the  results  will  be  equal.  We  choose,  then, 
the  least  quantity,  which  will  convert  all  the  fractions 
into  simple  quantities,  that  is,  the  least  common  mul- 
tiple of  the  denominators  3,  5,  10,  and  6,  which  is  30. 
If  we  multiply  both  equal  quantities  by  30,  the  equa- 
tion becomes 

30*       60*  _  210       30(3  —  2*) 

~3~       ~~5~      :~TO~~         ~~6~~ 

30*.    30  60*.    60 

But  -g-is-g-X*,  or  10*,  -g-  is-jr-X*,  or  12*,  etc.; 

so  that  we  have 

10*4-12*^21  —  5(3  —  2*),  (2) 

or  10*4-12*  =  21  —  (15  — 10*),  (3) 

or  10*-f  12*  =  21  — 15  +  10*.  (4) 

Beginners  very  commonly  mistake  this  process,  and 
forget  that  the  sign  of  subtraction,  when  it  is  written 
before  a  fraction,  implies  that  the  whole  result  of 
the  fraction  is  to  be  subtracted  from  the  rest.  As 
long  as  the  denominator  remains,  there  is  no  need  to 


EQUATIONS  OF  THE  FIRST  DEGREE.  99 

signify  this  by  putting  the  numerator  between  brack- 
ets, but  when  the  denominator  is  taken  away,  unless 
this  be  done,  the  sign  of  subtraction  belongs  to  the 
first  term  of  the  numerator  only,  and  not  to  the  whole 
expression.  The  way  to  avoid  this  mistake  would  be 
to  place  in  brackets  the  numerators  of  all  fractions 
which  have  the  negative  sign  before  them,  and  not  to 
remove  those  brackets  until  the  operation  of  subtrac- 
tion has  been  performed,  as  is  done  in  equation  (4). 
The  following  operations  will  afford  exercise  to  the 
student,  sufficient,  perhaps,  to  enable  him  to  avoid 
this  error  : 

«+-         f  f  ,.. 

a ^ = 7. — 


a  +  b      r+ 
We  can  now  proceed  with  the  solution  of  the  equa- 
tion.    Taking  up  the  equation  (4)  which  we  have  de- 
duced from  it,  subtract  10  x  from  both  sides,  which 
gives  10*  +  12*  —  10^  =  21  —  15,  or  12^  =  6:  divide 

these  equal  quantities  by  12,  which  gives  -r-^-  =^-^,  or 

1  1          U 

x  =  J.  This  is  the  only  value  which  x  can  have  so  as 
to  make  the  given  equation  true,  or,  as  it  is  called,  to 
satisfy  the  equation.  If  instead  of  x  we  substitute  ^, 
we  shall  find  that 


IOO  ON  THE  STUDY  OF  MATHEMATICS. 


j.^          ~         (r          __  . 

3  5       ~10  6  6  ^    5  ~10    ~  6' 

this  we  find  to  be  true,  since 

1         1         11          ,    7         2        22         ,  11        22 
¥+5    1S30'  and!0-  6-==66'  and  30  =  60- 
In  these  equations  of  the  first  degree  there  is  one  un- 
known quantity  and  all  the  others  are  known.     These 
known  quantities  may  be  represented  by  letters,  and, 
as  we  have  said,  the  first  letters  of  the  alphabet  are 
commonly  used  for  that  purpose.     We  will  now  take 
an  equation  of  exactly  the  same  form  as  the  last,  put- 
ting letters  in  place  of  numbers  : 

£L  _i_  —  —  —  __  -^H££ 

a  H      c   ~~  =  e  "          h 

The  solution  of  this  equation  is  as  follows  :  multi- 
ply both  quantities  by  aceh,  the  most  simple  multiple 
of  the  denominators,  it  then  becomes  : 

acehx       abcehx       acdeh        aceh(f  —  goc) 
~~~  ~~  ~~~  ~~     ~' 


or,   cehx-\-  ab  ehx  =  acdh  —  ace(f  —  gx), 
or,   cehx-\-abehx  =  acdh  —  acef-\-  acegx. 
Subtract  acegx  from  both  sides,  and  it  becomes 

cehx  -j-  abehx  —  acegx=acdh  —  acef, 
or,   (ceh-\-  ab  eh  —  aceg}x  =  acdh  —  ac  ef. 
Divide  both  sides  byceh  +  abeh  —  aceg,  which  gives 
ac  dh  —  acef 

*r  -  _  1  _ 

ceh-\-  abeh  —  ac  eg 

The  steps  of  the  process  in  the  second  case  are  ex- 
actly the  same  as  in  the  first  ;  the  same  reasoning  es- 


EQUATIONS  OF  THE  FIRST  DEGREE.  IOI 

tablishes  them  both,  and  the  same  errors  are  to  be 
avoided  in  each.     If  from  this  we  wish  to  find  the  so- 
lution of  the  equation  first  given,  we  must  substitute 
3  for  a,  2  for  b,  5  for  c,  7  for  d,  10  for  <?,  3  for  /,  2  for 
g,  and  6  for  h,  which  gives  for  the  value  of  x, 
3X5X7X6  —  3X5X10X3 
3X2X10X6  —  3X5 
3x5x12  180 


'   3X2X10X6'       '    360' 
which  is  £,  the  same  as  before. 

If  in  one  equation  there  are  two  unknown  quanti- 
ties, the  condition  is  not  sufficient  to  fix  the  values  of 
the  two  quantities  ;  it  connects  them,  nevertheless,  so 
that  if  one  can  be  found  the  other  can  be  found  also. 
For  example,  the  equation  x  -\-  y  =  S  admits  of  an  in- 
finite number  of  solutions,  for  take  x  to  represent  any 
whole  number  or  fraction  less  than  8,  and  let  y  repre- 
sent what  x  wants  of  8,  and  this  equation  is  satisfied. 
If  we  have  another  equation  of  condition  existing  be- 
tween the  same  quantities,  for  example,  3x  —  2y  =  4', 
this  second  equation  by  itself  has  an  infinite  number 

of  solutions:  to  find  them,  y  may  be  taken  at  pleasure, 

44-2v 
and  x  =  —  ~-  -.    Of  all  the  solutions  of  the  second 

D 

equation,  one  only  is  a  solution  of  the  first  ;  thus  there 
is  only  one  value  of  x  and  y  which  satisfies  both  the 
equations,  and  the  finding  of  these  values  is  the  solu- 
tion of  the  equations.  But  there  are  some  particular 
cases  in  which  every  value  of  x  and  y  which  satisfies 
one  of  the  equations  satisfies  the  other  also;  this  hap- 


IO2  ON  THE  STUDY  OF  MATHEMATICS. 

pens  whenever  one  of  the  equations  can  be  deduced 
from  the  other.  For  example,  when  x-\-y  =  8,  and 
4x  —  29  =  3  —  4j,  the  second  of  these  is  the  same  as 
4^  +  47  =  3  +  29,  or  4*  +  4jF  =  32,  which  necessarily 
follows  from  the  first  equation. 

If  the  solution  of  a  problem  should  lead  to  two 
equations  of  this  sort,  it  is  a  sign  that  the  problem 
admits  of  an  infinite  number  of  solutions,  or  is  what 
is  called  an  indeterminate  problem.  The  solution  of 
equations  of  the  first  degree  does  not  contain  any  pe- 
culiar difficulty ;  we  shall  therefore  proceed  to  the 
consideration  of  the  isolated  negative  sign. 


CHAPTER  IX. 

ON  THE  NEGATIVE  SIGN,  ETC. 

IF  we  wish  to  say  that  8  is  greater  than  5  by  the 
number  3,  we  write  this  equation  8 — 5  =  3.  Also 
to  say  that  a  exceeds  b  by  c,  we  use  the  equation  a — b 
=  c.  As  long  as  some  numbers  whose  value  we  know 
are  subtracted  from  others  equally  known,  there  is  no 
fear  of  our  attempting  to  subtract  the  greater  from 
the  less;  of  our  writing  3  —  8,  for  example,  instead  of 
8  —  3.  But  in  prosecuting  investigations  in  which  let- 
ters occur,  we  are  liable,  sometimes  from  inattention, 
sometimes  from  ignorance  as  to  which  is  the  greater 
of  two  quantities,  or  from  misconception  of  some  of 
the  conditions  of  a  problem,  to  reverse  the  quantities 
in  a  subtraction,  for  example  to  write  a  —  b  where  b 
is  the  greater  of  two  quantities,  instead  of  b  —  a.  Had 
we  done  this  with  the  sum  of  two  quantities,  it  would 
have  made  no  difference,  because  a  -f  b  and  b  +  a  are 
the  same,  but  this  is  not  the  case  with  a  —  b  and  b  —  a. 
For  example,  8  —  3  is  easily  understood;  3  can  be 
taken  from  8  and  the  remainder  is  5 ;  but  3  —  8  is  an 


104  ON  THE  STUDY  OF  MATHEMATICS. 

impossibility,  it  requires  you  to  take  from  3  more  than 
there  is  in  3,  which  is  absurd.  If  such  an  expression 
as  3  —  8  should  be  the  answer  to  a  problem,  it  would 
denote  either  that  there  was  some  absurdity  inherent 
in  the  problem  itself,  or  in  the  manner  of  putting  it 
into  an  equation.  Nevertheless,  as  such  answers  will 
occur,  the  student  must  be  aware  what  sort  of  mis- 
takes give  rise  to  them,  and  in  what  manner  they  af- 
fect the  process  of  investigation. 

We  would  recommend  to  the  beginner  to  make 
experience  his  only  guide  in  forming  his  notions  of 
these  quantities,  that  is,  to  draw  his  rules  from  the 
observation  of  many  results,  not  from  any  theory. 
The  difficulties  which  encompass  the  theory  of  the 
negative  sign  are  explained  at  best  in  a  manner  which 
would  embarrass  him  :  probably  he  would  not  see  the 
difficulties  themselves ;  too  easy  belief  has  always 
been  the  fault  of  young  students  in  mathematics,  and 
it  is  a  great  point  gained  to  get  them  to  start  an  ob- 
jection. We  shall  observe  the  effect  of  this  error  in 
denoting  a  subtraction  on  every  species  of  investiga- 
tion to  which  we  have  hitherto  come,  and  shall  de- 
duce rules  which  the  student  will  recollect  are  the  re- 
sults of  experience,  not  of  abstract  reasoning.  The 
extensions  to  which  he  will  be  led  have  rendered  Al- 
gebra much  more  general  than  it  was  before,  have 
made  it  competent  to  the  solution  of  many,  very  many 
questions  which  it  could  not  have  touched  had  they 
not  been  attended  to.  They  do,  in  fact,  constitute 


ON  THE  NEGATIVE  SIGN,   ETC.  105 

part  of  the  groundwork  of  modern  Algebra  and  should 
be  considered  by  the  student  who  is  desirous  of  mak- 
ing his  way  into  the  depths  of  the  science  with  the 
highest  degree  of  attention.  If  he  is  well  practised  in 
the  ordinary  rules  which  have  hitherto  been  explained, 
few  difficulties  can  afterwards  embarrass  him,  except 
those  which  arise  from  some  confusion  in  the  notions 
which  he  has  formed  upon  this  part  of  the  subject. 

For  brevity's  sake  we  hereafter  use  this  phrase. 
Where  the  signs  of  every  term  in  an  expression  are 
changed,  it  is  said  to  have  changed  its  form.  Thus 
-\- a  —  b  and  +  b  —  a  are  in  different  forms,  and  if  a 
be  greater  than  b,  the  first  is  the  correct  form  and  the 
second  incorrect.  An  extension  of  a  rule  is  made  by 
which  such  a  quantity  as  3  —  8  is  written  in  a  different 
way.  Suppose  that  -f-3 — 8  is  connected  with  any 
other  number  thus,  56  -j-  3  —  8.  This  may  be  written 
56  +  3— (3  +  5),  or  56  +  3  —  3  —  5,  or  56  — 5.  It  ap- 
pears, then,  that  +3 — 8,  connected  with  any  number 
is  the  same  as  — 5  connected  with  that  number;  from 
this  we  say  that  +3  —  8,  or  3  — 8  is  the  same  thing 
as  — 5,  or  3  —  8  =  —  5.  This  is  another  way  of  writ- 
ing the  equation  8  —  3  —  5,  and  indicates  equally  that 
8  is  greater  than  5  by  3.  In  the  same  way,  a  —  b  = 
—  c  indicates  that  b  is  greater  than  a  by  the  quantity 
c.  If  a  be  nothing,  this  equation  becomes  — b  =  —  c, 
which  indicates  that  b  =  c,  since  if  the  equation  a  —  b 
=  —  c  be  written  in  its  true  form  b  —  a  =  c^  and  if 


106  ON  THE  STUDY  OF  MATHEMATICS. 

0  =  0,  then  b  =  c.   We  can  now  understand  the  follow- 
ing equations  : 

a  —  b -f-  c — d=  —  e,  or  b-\-d — a  —  c  =  e, 
2ab  —  a?  —  b'*  =  —  d—e,  or  a2  +  P  —  2ab  =  d+  e. 

We  must  not  commence  any  operations  upon  such 
an  equation  as  a  —  b  = —  c,  until  we  have  satisfied  our- 
selves of  the  manner  in  which  they  should  be  per- 
formed, by  reference  to  the  correct  form  of  the  equa- 
tion. This  correct  form  is  b — a  =  c.  This  gives 
d-\-b — a  =  d-{-c,  or  d — {a — b~}  =  d-}-c.  Write  in- 
stead of  a — b  its  symbol — cy  and  then  d — ( — r)  = 
d-\-  c.  Here  we  have  performed  an  operation  with 
a  —  b,  which  is  no  quantity,  since  a  is  less  than  b,  but 
this  is  done  because  our  present  object  is,  in  applying 
the  common  rules  to  such  expressions,  to  watch  the 
results  and  exhibit  them  in  their  real  forms.  The  first 
side  d — ( — <r)  is  in  a  form  in  which  we  can  attach  no 
meaning  to  it,  and  the  second  side  gives  its  real  form 
d-\-  c.  The  meaning  of  this  expression  is,  that  if  with 
a  —  b,  which  we  think  to  be  a  quantity,  but  which  is 
not,  since  a  is  less  than  b,  we  follow  the  algebraical 
rule  in  subtracting  a  —  b  from  d,  we  shall  thereby  get 
the  same  result  as  if  we  had  added  the  real  quantity 
b  —  a  to  d.  If  we  make  use  of  the  form  d — ( — r),  it 
is  because  we  can  use  it  in  such  a  manner  as  never  to 
lose  sight  of  its  connexion  with  its  real  form  d -\- c, 
and  because  we  can  establish  rules  which  will  lead  us 
to  the  end  of  a  process  without  any  error,  except  those 


ON  THE  NEGATIVE  SIGN,  ETC.  107 

which  we  can  correct  as  certainly  at  the  end  as  at  the 
beginning. 

The  rule  by  which  we  proceed,  and  which  we  shall 
establish  by  numerous  examples,  is,  that  wherever 
two  like  signs  come  together,  the  corresponding  part 
of  the  real  form  has  a  positive  sign,  and  wherever  two 
unlike  signs  come  together,  the  real  form  has  a  nega- 
tive sign.  Thus  the  real  form  of  d — ( —  c]  is  d-{-c. 
Again,  take  the  real  form  b  —  a  =  c  of  the  equation 
a  —  &  =  —  c,  and  it  follows  that  d — (b  —  a)  =  d — c, 
or  d — b-\-a  =  d — c,  or  d  -f-  a  —  b  =  d — c,  or  d -\- 
(a — b}  =  d — c.  This  is  d-\-  (  —  *:)  —  */ — c,  another 
case  in  which  the  rule  is  verified.  Again,  multiply 
together  a  —  b  and  m  —  n,  the  product  is  am  —  an  — 
bm-\-bn.  This  is  the  same  product  as  arises  from 
multiplying  b  —  a  by  n  —  m,  written  in  a  different  or- 
der. If,  then,  b  —  a  =  c,  and  n — m=p,  or  a  —  6  = 
—  c,  and  m  —  ;*  —  — /,  we  find  that  ( — c)  X  ( — P)  = 
cp.  By  which  result  we  mean  that  a  mistake,  in  the 
form  of  both  a  —  b  and  m  —  n,  will  not  produce  a  mis- 
take in  the  form  of  their  product,  which  remains  what 
it  would  have  been  had  the  mistake  not  been  made. 
Again 

(n  —  ni)(b —  a~]  =  bn  —  bm  —  an -{-am 
(n  —  m}(a  —  b}  =  an  —  am  —  b  n  -f-  b  m. 

If  the  first  product  be  real  and  equal  to  P,  the  second 
is  represented  by  — P.  The  first  is  cp,  the  second  is 
( — <OXA  which  gives 


108  ON  THE  STUDY  OF  MATHEMATICS. 


That  is,  a  mistake  in  the  form  of  one  factor  only  alters 
the  form  of  the  product.  To  distinguish  the  right 
form  from  the  wrong  one,  we  may  prefix  -f-  to  the 
first,  and  —  to  the  second,  and  we  may  then  recapit- 
ulate the  results,  and  add  others,  which  the  student 
will  now  be  able  to  verify. 

The  sign  -f-  placed  before  single  quantities  shows 
that  the  form  of  the  quantity  is  correct  ;  the  sign  - 
shows  that  it  has  been  mistaken  or  changed. 


(+«)X  (+£)  =  +  «*         (+*)X(—  £)  =  —  ab 
(—  *)X(—  *)  =  +  «*  =  (+  «) 


.      ^ 
~     h  b 


-}-  a 


etc.  etc. 

We  see,  then,  that  a  change  in  the  form  of  any 
quantity  changes  the  form  of  those  powers  whose  ex- 
ponent is  an  odd  number,  but  not  of  those  whose  ex- 
ponent is  an  even  number.  By  these  rules  we  shall 


ON  THE  NEGATIVE  StGN,   ETC.  IOC) 

be  able  to  tell  what  changes  would  be  made  in  an  ex- 
pression by  altering  the  forms  of  any  of  its  letters.  It 
may  be  fairly  asked  whether  we  are  not  changing  the 
meaning  of  the  signs  -f  and  — ,  in  making  -f  a  stand 
for  an  expression  in  which  we  do  not  alter  the  signs, 
and  — a  for  one  in  which  the  signs  are  altered.  The 
change  is  only  in  name,  for  since  the  rule  of  addition 
is,  "  annex  the  expressions  which  are  to  be  added 
without  altering  the  signs  of  either,"  or  " annex  the 
expressions  without  altering  the  form  of  either  ;  "  the 
quantity  a  -j-  b,  which  is  the  sum  of  the  two  expres- 
sions a  and  b,  stands  for  the  same  as  -\-a-\-b,  in 
which  the  new  notion  of  the  sign  -j-  is  used,  viz.,  the 
expressions  a  and  b  are  annexed  with  unaltered  forms, 
which  is  denoted  by  writing  together  -f  a  and  -f  b. 
Again,  the  rule  for  subtraction  is,  "  change  the  sign 
of  the  subtrahend  or  expression  which  is  to  be  sub- 
tracted, and  annex  the  result  to  the  other  expression," 
or  "change  the  form  of  the  subtrahend  and  annex  it 
to  the  other,  which,  the  expressions  being  a  and  b,  is 
written  a  —  b,  which  answers  equally  well  to  the  sec- 
ond notion  of  the  sign  — ,  since  -\-a — b  indicates  that 
a  and  b  are  to  be  annexed,  the  first  without,  the  sec- 
ond with  a  change  of  form.  These  ideas  of  the  signs 
-f  and  —  give,  therefore,  in  practice,  the  same  results 
as  the  former  ones,  and,  in  future,  the  two  meanings 
may  be  used  indiscriminately.  But  when  a  single 
term  is  used,  such  as  -\-  a  or  — a,  the  last  acquired 
notions  of  -f-  and  —  are  always  understood. 


IIO  ON  THE  STUDY  OF  MATHEMATICS. 

This  much  being  premised,  we  can  see,  by  num- 
berless instances,  that,  if  the  form  of  a  quantity  is  to 
be  changed,  it  matters  nothing  whether  it  is  changed 
at  the  beginning  of  the  process,  or  whether  we  wait 
till  the  end,  and  then  follow  the  rules  above  men- 
tioned. This  is  evident  to  the  more  advanced  stu- 
dent, from  the  nature  of  the  rules  themselves,  but  the 
beginner  should  satisfy  himself  of  this  fact  from  expe- 
rience. We  now  give  a  proof  of  this,  as  far  as  one 
expression  can  prove  it,  in  the  solution  of  the  equa- 
tions, 


.   a2  a?  x 

and  -  --  ax  =  —  -  --  a 


which  two  equations  only  differ  in  the  form  in  which 
a  appears.     For,  if  the  form  of  a  in  the  first  equation 

be  altered,  that  of—  and  —  —  is  unaltered,  -{-ax  be- 
b  b 

comes  —  ax,  and  -\-  a  becomes  —  a.     We  now  solve 
the  two  equations  in  opposite  columns. 


J+ax=-_+a_t,  __«*;=__,,_, 


a2-\-atx  =  a<2  x  +  ab  —  ti*        a2  —  atx  =  a'2x  —  ab  — 

a*      ab      P  =  a>  x      abx 


_  _ 


The    only   difference    between   these    expressions 


ON  THE  NEGATIVE  SIGN,  ETC.  Ill 

arises  from  the  different  form  of  a  in  the  two.  If,  in 
either  of  them,  —  a  be  put  instead  of  -f  a,  and  the 
rules  laid  down  be  followed,  the  other  will  be  pro- 
duced. We  see,  then,  that  a  simple  alteration  of  the 
form  of  a  in  the  original  equation  produces  no  other 
change  in  the  result,  or  in  any  one  of  the  steps  which 
lead  to  that  result,  except  a  simple  alteration  in  the 
form  of  a.  From  this  it  follows  that,  having  the  so- 
lution of  an  equation,  we  have  also  the  solution  of  all 
the  equations  which  can  be  formed  from  it,  by  altering 
the  form  of  the  different  known  quantities  which  are 
contained  in  it.  And,  as  all  problems  can  be  reduced 
to  equations,  the  solution  of  one  problem  will  lead  us 
to  the  solution  of  others,  which  differ  from  the  first  in 
producing  equations  in  which  some  of  the  known 
quantities  are  in  different  forms.  Also,  in  every  iden- 
tical equation,  the  form  of  one  or  more  of  its  quanti- 
ties may  be  altered  throughout,  and  the  equation  will 
still  remain  identically  true.  For  example, 


a  —  o 
Change  -f  b  into  —  1>,  and  this  equation  will  become 


which  last,  common  division  will  show  to  be  true. 

Again,  suppose  than  when  a,  b,  and  c  are  in  a 
given  form,  which  we  denote  by  -f-  a,  -f-  b,  and  -f-  c, 
the  solution  of  a  problem  is, 


112  ON  THE  STUDY  OP  MATHEMATICS* 


The  following  table  will  show  the  alterations  which 
take  place  in  x  when  the  forms  of  a,  b,  and  c  are 
changed  in  different  manners,  and  the  verification  of 
it  will  be  an  exercise  for  the  student. 

FORMS  OF  a,  b,  AND  C.  VALUES  OF  X. 

+  a,+b,+c  ^T**^ 

a-\-  c — b 

+  «.+*.-'  a-c-t 

-\-a,  — bt  — c  — 7- 

a  —  c-\-b 


—  a,  — bt  — c 


b  —  a — c 


Also,  the  expression  for  x  may  be  written  in  the 
following  different  ways,  the  forms  of  a,  b,  and  c  re- 
maining the  same  : 


a  +  c  —  b'          b  —  a  —  c'          a-\-c  —  b1     b  —  a  —  c' 

We  now  proceed  to  apply  these  principles  to  the 
solution  of  the  following  problems  : 

q  -  1  ---  1  --  j  -  \D 

A  B  H 

Two  couriers,  A  and  B,  in  the  course  of  a  journey 
between  the  towns  C  and  D,  are  at  the  same  moment 


ON  THE  NEGATIVE  SIGN,   ETC.  113 

of  time  at  A  and  B.  A  goes  m  miles,  and  B,  n  miles 
an  hour.  At  what  point  between  C  and  D  are  they 
together  ?  It  is  evident  that  the  answer  depends  upon 
whether  they  are  going  in  the  same  or  opposite  direc- 
tions, whether  A  goes  faster  or  slower  than  B,  and  so 
on.  But  all  these,  as  we  shall  see,  are  included  in 
the  same  general  problem,  the  difference  between  them 
corresponding  to  the  different  forms  of  the  letters 
which  we  shall  have  occasion  to  use.  After  solving 
the  different  cases  which  present  themselves,  each 
upon  its  own  principle,  we  shall  compare  the  results 
in  order  to  establish  their  connexion.  Let  the  dis- 
tance AB  be  called  a. 

Case  first. — Suppose  that  they  are  going  in  the 
same  direction  from  C  to  D,  and  that  m  is  greater  than 
n.  They  will  then  meet  at  some  point  between  B  and 
D.  Let  that  point  be  H,  and  let  AH  be  called  x. 
Then  A  travels  through  AH,  or  x,  in  the  time  during 
which  B  travels  through  BH,  or  x — a.  But,  since  A 
goes  m  miles  an  hour,  he  travels  the  distance  x  in 

—  hours.    Again,  B  travels  the  distance  x — a  in  - 
m  n 

hours.     These  times  are  the  same,  and,  therefore, 

x        x  —  a  ma 

—  — or  x  —  -      -  =  AH 

m  n  m  —  n 


m — n 
The  time  which  elapses  before  they  meet  is 

x            a 
—  or . 


in        m  • 


114  ON  THE  STUDY  OF  MATHEMATICS. 

Case  second.  —  Suppose  them  now  moving  in  the 
same  direction  as  before,  but  let  B  move  faster  than 

A.  They  never  will  meet  after  they  come  to  A  and 

B,  since  B  is  continually  gaining  upon  A,  but  they 
must  have  met  at  some  point  before  reaching  A  and 
B.     Let  that  point  be  H,  and,  as  before,  let  AH=x. 

q  -  ^  --  J.  -  j  ---  \D          i 

Then  since  A  travels  through  HA  or  x  in  the  time 
during  which  B  travels  through  HB,  or  x  -\-  a,  in  the 
same  manner  as  in  the  last  case,  we  show  that 

x        x  -\-  a  ma 

-=-     -or*—  -      -  —  AH 
m  n  n  —  m 


-  =BH. 

;;/ 


The  time  elapsed  is  ... 


m 


Case  third.  —  If  they  are  moving  from  D  to  C,  and 
if  B  moves  faster  than  A,  the  point  His  the  same  as 
in  the  last  case,  since,  if  having  in  the  last  case  ar- 
rived at  A  and  B,  they  move  back  again  at  the  same 
rate,  they  will  both  arrive  at  the  point  H  together. 
The  answers  in  this  case  are  therefore  the  same  as  in 
the  last. 

Case  fourth.—  Similarly,  if  they  are  moving  from  D 
to  C,  and  A  moves  faster  than  B,  the  answers  are  the 
same  as  in  the  first  case,  since  this  is  a  reverse  of  the 
first  case,  as  the  third  is  of  the  second.  We  reserve 


ON  THE  NEGATIVE  SIGN,   ETC.  115 

for  the  present  the  case  in  which  they  move  equally 
fast,  as  another  species  of  difficulty  is  involved  which 
has  no  connexion  with  the  present  subject.  We  shall 
return  to  it  hereafter. 

Case  fifth. — Suppose  them  now  moving  in  contrary 
directions,  viz.:  A  towards  D  and  B  towards  C. 
Whether  A  moves  faster  or  slower  than  JB,  they  must 
now  meet  somewhere  between  A  and  B\  as  before  let 
them  meet  in  H9  and  let  Aff=x. 

C\ +_, J. 1* 

Then  A  moves  through  AH,  or  x,  in  the  same  time  as 
B  moves  through  BH,  or  a  —  x.  Therefore 

x        a  —  x 

—  = ,  or 

m  n 

ma 

x  = 


a  —  x  = 


The  time  elapsed  is .  .  . 


m-\-n 

na 

m  -\-  n 

a 
m-\-  n 


Case  sixth. — Let  them  be  moving  in  contrary  direc- 
tions, but  let  A  be  moving  towards  C,  and  B  towards 
D.  They  will  then  have  met  somewhere  between  A 
and  Bt  and  as  this  is  only  the  reverse  of  the  last  case, 
just  as  the  fourth  is  of  the  first,  or  the  third  of  the 
second,  the  answers  are  the  same.  We  now  exhibit 
the  results  of  these  different  cases  in  a  table,  stating 


n6 


ON  THE  STUDY  OF  MATHEMATICS. 


the  circumstances  of  each  case,  and  also  whether  the 
time  of  meeting  is  before  or  after  the  instant  which 
finds  them  at  A  and  B. 


Circumstances  of  the  case. 

Direction  of 
the  point  H. 

Value 
of  AH. 

Value 
of  BH. 

Time  of  meeting 

j  Both  move  from  Cto  D, 

Between 

ma 

n  a 

a        -iftrr 

t  A  moves  faster  than  B. 
(  Both  move  from  Cto  D, 

B  and  D. 
Between 

m  —  » 
nta 

•m  —  n 
n  a 

m  —  n 
a        hrfnrr 

i  A  moves  slower  than  B. 
j  Both  move  from  D  to  C, 

A  and  C. 
Between 

n  —  m 
ma 

n  —  m 
na 

n-mbCfOTC- 

/r 
after 

5'    '  A  moves  slower  than  B. 
(  Both  move  from  D  to  C, 

A  and  C. 
Between 

n  —  m 
ma 

n  —  m 
n  a 

n  —  m 

<  A  moves  faster  than  B. 
j  A  moves  towards  D  and 

B  and  D. 
Between 

m  —  n 
ma 

m  —  n 
na 

m  —  n 
a         tt 

5'   1  B  towards  C 
j  A  moves  towards  C  and 

A  and  B. 
Between 

m  +  n 
m  a 

m  +  n 
n  a 

m  +  n 

a. 
hrfnrr 

J<    '  B  towards  D. 

A  and  B. 

m  -\-  n 

m  -\-  n 

m  +  n 

Now 


a 


and are  the  same  quantity  written  in 

m  —  n         n — m 

different  forms,  for  n  —  m  is  — (m  — «);  and  accord- 
ing to  the  rules 

a  a 


m 


m 


Similarly 


ma 


n — m  m  —  n 

and  so  on. 

We  see  also,  that  in  the  first  and  second  cases,  which 
differ  in  this,  that  AH  falls  to  the  right  in  the  first, 
and  to  the  left  in  the  second,  the  forms  of  AH are 

different,  there  being in  the  first,  and 

& 


ON  THE  NEGATIVE  SIGN,   ETC.  117 

in  the  second.  Again,  in  the  same  cases,  in  the  first 
of  which  the  time  of  meeting  is  after,  and  in  the  sec- 
ond before  the  moment  of  being  at  A  and  B,  we  see  a 
difference  of  form  in  the  value  of  that  time ;  in  the 
first  it  is ,  and  in  the  second ,  or 


m  —  n  m  —  n        n  —  m 

The  same  remarks  apply  to  the  third  and  fourth  ex- 
amples. Again,  in  the  first  and  fifth  cases,  which  only 
differ  in  this,  that  B  is  moving  towards  D  in  the  first, 
and  in  the  contrary  direction  towards  C  in  the  fifth, 
the  values  of  AH,  and  of  the  time,  may  be  deduced 
from  the  first  by  changing  the  form  of  «,  and  writing 
-(-  nt  instead  of  — n.  The  expression  for  BH  in  the 
first,  if  the  form  of  n  be  likewise  changed,  becomes 

,  which  is  the  value  of  BH\n  the  fifth,  but  in 

m  -f  n 
a  different  form.    But  we  observe  that  BH  falls  to  the 

left  of  B  in  the  fifth,  whereas  it  fell  to  the  right  in  the 
first.  Again,  in  the  first  and  sixth  examples,  which 
differ  in  this  that  A  moves  towards  D  in  the  first  and 
towards  C  in  the  sixth,  the  value  of  AH  in  the  sixth 
may  be  deduced  from  that  of  AH  in  the  first  by 

changing  the  form  of  m,  which  change  makes  AH  be- 

• —  ma  —  ma  ma 

come — ,  or  — 7 — - — - ,  or  — ; — .   If  we  alter  the 

—  m — n        — (m-\-n)        m-\-n 

value  of  the  time  in  the  first,  in  the  same  manner,  it 

becomes ,  or — ,  which  is  of  a  different 

—  m  —  n  m-\-n 

form  from  that  in  the  sixth ;  but  it  must  also  be  ob- 
served that  the  first  is  after  and  the  other  before  the 
moment  when  they  are  at  A  and  B.  In  the  fifth  and 
sixth  examples  which  differ  in  this,  that  the  direction 


n8 


ON  THE  STUDY  OF  MATHEMATICS. 


in  which  both  are  going  is  changed,  since  in  the  fifth 
they  move  towards  one  another,  and  in  the  sixth  away 
from  one  another,  the  values  of  AH  and  BH  in  the 
one  may  be  deduced  from  those  in  the  other  by  a 
change  of  form,  both  in  m  and  n,  which  gives  the 
same  values  as  before.  But  if  m  and  n  change  their 
forms  in  the  expression  for  the  time,  the  value  in  the 


sixth  case  is 


-,  or  — 


m-\-  n 


Also   the  time  in 


Circumstances  of  the  case. 

Direction  of 
the  point  H. 

Value 
of  AH. 

Value 
of  BH. 

Time  of  meeting 

j  Both  move  from  C  to  D, 

Between 

ma 

n  a 

a        iftpr 

1  A  moves  faster  than  B. 
j  Both  move  from  C  to  Z>, 

B  and  D, 
Between 

m  —  n 
ma 

m  —  n 

n  a 

m  —  n 

'  A  moves  slower  than  B. 
j  Both  move  from  D  to  C, 

A  and  C. 
Between 

n  —  tn 
ma 

n  —  m 
na 

n  —  m 

a         ft 

3'   1  A  moves  slower  than  B. 
j  Both  move  from  D  to  C, 

A  and  C. 
Between 

n  —  m 
ma 

n  —  m 
na 

n  —  m  a  tCr' 

(A  moves  faster  than  B. 
j  A  moves  towards  D  and 

B  and  D. 
Between 

m  —  » 
ma 

m  —  n 
na 

m  —  n 
a      after 

5<   1  B  towards  C. 
j  A  moves  towards  C  and 

A  and  B. 
Between 

m  +  n 
ma 

m  +  n 
na 

m  +  » 

'*   1  B  towards  D. 

A  and  B. 

m  -\-  n 

m  +  n 

m  +  n 

(TABLE  OF  PAGE  116  REPEATED.) 

the  fifth  case  is  after  the  moment  at  which  they  are 
at  A  and  B,  and  in  the  sixth  case  it  is  before.  From 
these  comparisons  we  deduce  the  following  general 
conclusions : 

1.  If  we  take  the  first  case  as  a  standard,  we  may, 
from  the  values  which  it  gives,  deduce  those  which 
hold  good  in  all  the  other  cases.  If  a  second  case  be 
taken,  and  it  is  required  to  deduce  answers  to  the 


ON  THE  NEGATIVE  SIGN,  ETC.  Iig 

second  case  from  those  of  the  first,  this  is  done  by 
changing  the  sign  of  all  those  quantities  whose  direc- 
tions are  opposite  in  the  second  case  to  what  they  are 
in  the  first,  and  if  any  answer  should  appear  in  a  neg- 
ative form,  such  as ,  when  m  is  less  than  », 

m — n 

which  may  be  written  thus ,  it  is  a  sign  that 

n — m 

the  quantity  which  it  represents  is  different  in  direc- 
tion in  the  first  and  second  cases.  If  it  be  a  right 
line  measured  from  a  given  point  in  all  the  cases, 
such  as  AH,  it  is  a  sign  that  AH  falls  on  the  left  in 
the  second  case,  if  it  fell  on  the  right  in  the  first  case, 
and  the  converse.  If  it  be  the  time  elapsed  between 
the  moment  in  which  the  couriers  are  at  A  and  B  and 
their  meeting,  it  is  a  sign  that  the  moment  of  meeting 
is  before  the  other,  in  the  second  case,  if  it  were  after 
it  in  the  first,  and  the  converse.  We  see,  then,  that 
these  six  cases  can  be  all  contained  in  one  if  we  apply 
this  rule,  and  it  is  indifferent  which  of  the  cases  is 
taken  as  the  standard,  provided  the  corresponding 
alterations  are  made  to  determine  answers  to  the  rest. 

This  detail  has  been  entered  into  in  order  that  the 
student  may  establish  from  his  own  experience  the 
general  principle  which  will  conclude  this  part  of  the 
subject.  Further  illustration  is  contained  in  the  fol- 
lowing problem : 

A  workman  receives  a  shillings  a  day  for  his  labor 
or  a  proportion  of  a  shillings  for  any  part  of  a  day 
which  he  works.  His  expenses  are  b  shillings  every 


120  ON  THE  STUDY  OF  MATHEMATICS. 

day,  whether  he  works  or  no,  and  after  m  days  he 
finds  that  he  has  gained  c  shillings.  How  many  days 
did  he  work?  Let  x  be  that  number  of  days,  x  being 
either  whole  or  fractional ;  then  for  his  work  he  re- 
ceives ax  shillings,  and  during  the  m  days  his  expen- 
diture is  bm  shillings,  and  since  his  gain  is  the  differ- 
ence between  his  receipts  and  expenditure : 
ax — bm  =  c 

bm-}-  c 

or  x= 

a 

Now  suppose  that  he  had  worked  so  little  as  to  lose  c 
shillings  instead  of  gaining  anything.  The  equation 
from  which  x  is  derived  is  now 

bm  —  a  x  =  Cj 
which,  when  its  form  is  changed,  becomes 

ax  —  bm  =  —  c, 

an  equation  which  only  differs  from  the  former  in  hav- 
ing —  c  written  instead  of  c.  The  solution  of  the  equa- 
tion is 

bm  —  c 

*=__, 

which  only  differs  from  the  former  in  having  — c  in- 
stead of  -|-  c.  It  appears  then  that  we  may  alter  the 
solution  of  a  problem  which  proceeds  upon  the  sup- 
position of  a  gain  into  the  solution  of  one  which  sup- 
poses an  equal  loss,  by  changing  the  form  of  the  ex- 
pression which  represents  that  gain ;  and  also  that  if 
the  answer  to  a  problem  which  we  have  solved  upon 
the  supposition  of  a  gain  should  happen  to  be  nega- 


ON  THE  NEGATIVE  SIGN,  ETC.  121 

tive,  suppose  it  — c,  we  should  have  proceeded  upon 
the  supposition  that  there  is  a  loss  and  should  in  that 
case  have  found  a  loss,  c.  When  such  principles  as 
these  have  been  established,  we  have  no  occasion  to 
correct  an  erroneous  solution  by  recommencing  the 
whole  process,  but  we  may,  by  means  of  the  form  of 
the  answer,  set  the  matter  right  at  the  end.  The 
principle  is,  that  a  negative  solution  indicates  that 
the  nature  of  the  answer  is  the  very  reverse  of  that 
which  it  was  supposed  to  be  in  the  solution ;  for  ex- 
ample, if  the  solution  supposes  a  line  measured  in 
feet  in  one  direction,  a  negative  answer,  such  as  — c, 
indicates  that  c  feet  must  be  measured  in  the  opposite 
direction ;  if  the  answer  was  thought  to  be  a  number 
of  days  after  a  certain  epoch,  the  solution  shows  that 
it  is  c  days  before  that  epoch ;  if  we  supposed  that  A 
was  to  receive  a  certain  number  of  pounds,  it  denotes 
that  he  is  to  pay  c  pounds,  and  so  on.  In  deducing 
this  principle  we  have  not  made  any  supposition  as 
to  what  —  c  is ;  we  have  not  asserted  that  it  indicates 
the  subtraction  of  c  from  0 ;  we  have  derived  the  re- 
sult from  observation  only,  which  taught  us  first  to 
deduce  rules  for  making  that  alteration  in  the  result 
which  arises  from  altering  -f  c  into — c  at  the  com- 
mencement ;  and  secondly,  how  to  make  the  solution 
of  one  case  of  a  problem  serve  to  determine  those  of 
all  the  others.  By  observation  then  the  student  must 
acquire  his  conviction  of  the  truth  of  these  rules,  re- 
serving all  metaphysical  discussion  upon  such  quanti- 


122  ON  THE  STUDY  OF  MATHEMATICS. 

ties  as  -|-  c  and  — c  to  a  later  stage,  when  he  will  be 
better  prepared  to  understand  the  difficulties  of  the 
subject.  We  now  proceed  to  another  class  of  difficul- 
ties, which  are  generally,  if  possible,  as  much  miscon- 
ceived by  the  beginner  as  the  use  of  the  negative  sign. 

Take  any  fraction  — .    Suppose  its  numerator  to 
b 

remain  the  same,  but  its  denominator  to  decrease,  by 
which  means  the  fraction  itself  is  increased.     For  ex- 
ample, ^  is  greater  than  ^r  or  the  twelfth  part  of  5 
14  ZU  2i 

is  greater  than  its  twentieth  part.     Similarly,  -  \  is 

2i  4i 

greater  than  g,  etc.     If,  then,  b  be  diminished  more 

and  more,  the  fraction  —  becomes  greater  and  greater, 
and  there  is  no  limit  to  its  possible  increase.  To  show 

this,  suppose  that  b  is  a  part  of  a,  or  that  b  =  — .  Then 

a         a  m 

-r-  or  —  is  m.     Now  since  b  may  diminish  so  as  to  be 

(M 

equal  to  any  part  of  a,  however  small,  that  is,  so  as 

to  make  m  any  number,  however  great,  —  which   is 

b 

=  m  may  be  any  number  however  great.  This  dimi- 
nution of  £,  and  the  consequent  increase  of—,  may  be 
carried  on  to  any  extent,  which  we  may  state  in  these 
words :  As  the  quantity  b  becomes  nearer  and  nearer 

to  0,  the  fraction  —  increases,  and  in  the  interval  in 
o 

which  b  passes  from  its  first  magnitude  to  0,  the  frac- 
tion —  passes  from  its  first  value  through  every  pos- 
sible greater  number.  Now,  suppose  that  the  solution 

of  a  problem  in  its  most  general  form  is  — ,  but  that 

b 


ON  THE  NEGATIVE  SIGN,   ETC.  12$ 

in  one  particular  case  of  that  problem  b  is  =  0.  We 
have  then  instead  of  a  solution  -^,  a  symbol  to  which 
we  have  not  hitherto  given  a  meaning. 

To  take  an  instance  :  return  to  the  problem  of  the 
two  couriers,  and  suppose  that  they  move  in  the  same 
direction  from  C  to  D  (Case  first']  at  the  same  rate,  or 

^1  TTT  £J^1  ,«    7-7-  ma  ma 

that  m  =  n.     We   find   that  Aff=  —  -  or  -    —  or 

ma  m — n       n  — n 

-j-.     On  looking  at  the  equation  which  produced  this 

xx a 

result  we  find  that  it  becomes  —  = ,  or  x=x — a, 

m  m 

which  is  impossible.  On  looking  at  the  manner  in 
which  this  equation  was  formed,  we  find  that  it  was 
made  on  the  supposition  that  A  and  B  are  together  at 
some  point,  which  in  this  case  is  also  impossible,  since 
if  they  move  at  the  same  rate,  the  same  distance  which 
separated  them  at  one  moment  will  separate  them  at 
any  other,  and  they  will  never  be  together,  nor  will 
they  ever  have  been  together  on  the  other  side  of  A. 
The  conclusion  to  be  drawn  is,  that  such  an  equation 
as  x  =  -^  indicates  that  the  supposition  from  which  x 
was  deduced  can  never  hold  good.  Nevertheless  in 
the  common  language  of  algebra  it  is  said  that  they 
meet  at  an  infinite  distance,  and  that  -~-  is  infinite. 
This  phrase  is  one  which  in  its  literal  meaning  is  an 
absurdity,  since  there  is  no  such  thing  as  an  infinite 
number,  that  is  a  number  which  is  greater  than  any 
other,  because  the  mind  can  set  no  bounds  to  the 
magnitude  of  the  numbers  which  it  can  conceive,  and 


124  ON  THE  STUDY  OF  MATHEMATICS. 

whatever  number  it  can  imagine,  however  great,  it 
can  imagine  the  next  to  it.  But  as  the  use  of  the 
phrase  is  very  general,  the  only  method  is  to  attach  a 
meaning  which  shall  not  involve  absurdity  or  con- 
fusion of  ideas.  The  phrase  used  is  this :  When 
c  =  b,  -  ——  =  —  and  is  infinitely  great.  The  student 
should  always  recollect  that  this  is  an  abbreviation  of 

the  following  sentence.    "The  fraction becomes 

c  —  o 

greater  and  greater  as  c  approaches  more  and  more 
near  to  b ;  and  if  c,  setting  out  from  a  certain  value, 
should  change  gradually  until  it  becomes  equal  to  b, 

the  fraction -j-  setting  out  also  from  a  certain  value, 

will  attain  any  magnitude  however  great,  before  c  be- 
comes equal  to  b."  That  is,  before  a  fraction  can  as- 
sume the  form  -^  ,  it  must  increase  without  limit.  The 
symbol  oo  is  used  to  denote  such  a  fraction,  or  in  gen- 
eral any  quantity  which  increases  without  limit.  The 
following  equation  will  tend  to  elucidate  the  use  of 

this  symbol.     In  the  problem  of  the  two  couriers,  the 

...  .,  ..  ma          x        x  —  a 

equation  which  gave  the  result  —^  was  —  =  —    — ,  or 

0  m  m 

x  =  x  —  a,  which  is  evidently  impossible.  Neverthe- 
less, the  larger  x  is  taken  the  more  near  is  this  equa- 
tion to  the  truth,  as  may  be  proved  by  dividing  both 

sides  by  x,  when  it  becomes  1  =  1—   — ,  which  is  never 

d      ^ 
exactly  true.     But  the  fraction  —  decreases   as  x  in- 

x 

creases,  and  by  taking  x  sufficiently  great  may  be  re- 
duced to  any  degree  of  smallness.  For  example,  if  it 


ON  THE  NEGATIVE  SIGN,   ETC.  125 

is  required  that  —  should  be  as  small  as  ^  of  a 

unit,  take  x  as  great  as  10000000  a,  and  the  fraction 

bec°mes  10000000 a'  °r  WOOO-     But  "^becomes 

smaller    and    smaller,    the    equation    1  =  1 —   -be- 

oc 

comes  nearer  and  nearer  the  truth,  which  is  expressed 
by  saying  that  when  1=1 ,  or  x=x  —  a,  the  so- 

OC 

lution  is  x  =  oo.  In  the  solution  of  the  problem  of 
the  two  couriers  this  does  not  appear  to  hold  good, 
since  when  m  =  n  and  x=--  the  same  distance  a 
always  separates  them,  and  no  travelling  will  bring 
them  nearer  together.  To  show  what  is  meant  by 
saying  that  the  greater  x  is,  the  nearer  will  it  be  a  so- 
lution of  the  problem,  suppose  them  to  have  travelled 
at  the  same  rate  to  a  great  distance  from  C.  They 


C\ 1 \B 

A 

can  never  come  together  unless  CA  becomes  equal  to 
CBt  or  A  coincides  with  B,  which  never  happens, 
since  the  distance  AB  is  always  the  same.  But  if  we 
suppose  that  they  have  met,  though  an  error  always 
will  arise  from  this  false  supposition,  it  will  become 
less  and  less  as  they  travel  farther  and  farther  from 
C.  For  example,  let  (7.4  =  10000000  AB,  then  the 
supposing  that  they  have  met,  or  that  B  and  A  co- 
incide, or  that  BA  =  0,  is  an  error  which  involves  no 
more  than  of  A  C;  and  though  AB  is  always 

of  the  same  numerical  magnitude,  it  grows  smaller 


126  ON  THE  STUDY  OF  MATHEMATICS. 

and  smaller  in  comparison  with  AC,  as  the  latter 
grows  greater  and  greater. 

Let  us  suppose  now  that  in  the  problem  of  the  two 
couriers  they  move  in  the  same  direction  at  the  same 
rate,  as  in  the  case  we  have  just  considered,  but  that 
moreover  they  set  out  from  the  same  point,  that  is, 
let  «  — 0.  It  is  now  evident  that  they. will  always  be 
together,  that  is,  that  any  value  of  x  whatever  is  an 
answer  to  the  question.  On  looking  at  the  value  of 

AH,  or ,  we  find  the  numerator  and  denomina- 

m — n 

tor  both  equal  to  0,  and  the  value  of  AH  appears  in 
the  form  -^.  But  from  the  problem  we  have  found 
that  one  value  cannot  be  assigned  to  AH,  since  every 
point  of  their  course  is  a  point  where  they  are  to- 
gether. The  solution  of  the  following  equation  will 
further  elucidate  this.  Let 

ax-\-  t>y  =  c 

dx  +  cy^f, 
from  which,  by  the  common  method  of  solution,  we 

find 

ce  —  bf  af — cd 

\ V 

—          ~i~j '    y  —          ~i~j 
ae — bd  ae — bd 

Now,  let  us  suppose  that  fe  =  ^fand  ae  =  bd.  Divid- 
ing the  first  of  these  by  the  second,  we  find 

ce       bf         c        f 
-  =  -^-,  or  —  ==  ^r,  or  cd=af. 

ae       bd         a        d 

The  values  both  of  x  and  y  in  this  case  assume  the 
form  -  to  find  the  cause  of  this  we  must  return  to 


ON  THE  NEGATIVE  SIGN,  ETC.  127 

the  equations.     If  we  divide  the  first  of  these  by  c, 
and  the  second  by/,  we  find  that 

a       .    b 
-*+-,=!. 


But  the  equations  <r^  =  £/and  r</=0/give  us  —  =  -, 
and  —  =  -j,  that  is,  these  two  are,  in  fact,  one  and 
the  same  equation  repeated,  from  which,  as  has  been 
explained  before,  an  infinite  number  of  values  of  x 
and  y  can  be  found  ;  in  fact,  any  value  may  be  given 
to  x  provided  y  be  then  found  from  the  equation.  We 
see  that  in  these  instances,  when  the  value  of  any 
quantity  appears  in  the  form  -^-,  that  quantity  admits 
of  an  infinite  number  of  values,  and  this  indicates  that 
the  conditions  given  to  determine  that  quantity  are 
not  sufficient.  But  this  is  not  the  only  cause  of  the 
appearance  of  a  fraction  in  the  form  -^-.  Take  the 
identical  equation 


When  a  approaches  towards  b,  a  -\-  b  approaches  to- 
wards 2  £,  and  a2  —  £2  and  a  —  b  approach  more  and 
more  nearly  towards  0.  If  a  =  b  the  equation  assumes 
this  form  : 


0 


128  ON  THE  STUDY  OP  MATHEMATICS. 

This  may  be  explained  thus :  if  we  multiply  the  nu- 

A 
merator  and  denominator  of  the  fraction  —   by  a  —  b 

J)  *  47 

(which  does  not  alter  its  value)  it  becomes  — ^7. 

Ha  —  no 

If  in  the  course  of   an  investigation  this  has  been 

done  when  the  two  quantities  a  and  b  are  equal  to 

A        Aa  —  Ab 

one  another,  the  fraction  -^  or  -=—  — —  will  appear 
A  ±>         ±>a  —  J3  o 

in  the  form  7-.     But  since  the  result  would  have  been 

^ 

—  had  that  multiplication  not  been  performed,  this 

JD 

last  fraction  must  be  used  instead  of  the  unmeaning 

.         0  a2  —  P         (a  +  b)(a  —  b) 

form  -JT-.     Thus  the  fraction  — —  or  - — '  , — - 

0  a  +  b  c(a  —  b^  c(a—b} 

is  the  fraction  -     -  after  its  numerator  and  denomi- 
c 

nator  have  been  multiplied  by  a — b,  and  may  be  used 
in  all  cases  except  that  in  which  a  —  b.  When  the 
form  —  occurs,  the  problem  must  be  carefully  ex- 
amined in  order  to  ascertain  the  reason. 


CHAPTER  X. 

EQUATIONS  OF  THE  SECOND  DEGREE. 

T^VERY  operation  of  algebra  is  connected  with  an- 
. 

-*-'    other  which  is  exactly  opposite  to  it  in  its  effects. 

Thus  addition  and  subtraction,  multiplication  and  di- 
vision, are  reverse  operations,  that  is,  what  is  done 
by  the  one  is  undone  by  the  other.  Thus  a  -f-  b  —  b  is 
0,  and  —  is  a.  Now  in  connexion  with  the  raising  of 
powers  is  a  contrary  operation  called  the  extraction 
of  roots.  The  term  root  is  thus  explained  :  We  have 
seen  that  a  a,  or  a2,  is  called  the  square  of  a  ;  from 
which  a  is  called  the  square  root  of  a2.  As  169  is 
called  the  square  of  13,  13  is  called  the  square  root  of 
169.  The  following  table  will  show  how  this  phrase- 
ology is  carried  on. 


a  is  called  the  square  root  of  a2,  .  .  denoted  by 
a  "  "  "  cube  root  of  «3,  .  .  "  "  ^~cP 
a  "  "  "  fourth  root  of  04,  .  .  "  "  \/~a*  '. 
a"  "  "  fifth  root  of  a5,  ..  "  "  v'~cP 
etc.  etc.  etc. 


13O  ON  THE  STUDY  OF  MATHEMATICS. 

If  b  stand  for  a5,  \/~b  stands  for  a,  and  the  foregoing 
table  may  be  represented  thus  : 

\ia*  =  b,a  =  Vb\ 

if  a*  =  b,a  =  $/~b,  etc. 

The  usual  method  of  proceeding  is  to  teach  the 
student  to  extract  the  square  root  of  any  algebraical 
quantity  immediately  after  the  solution  of  equations 
of  the  first  degree.  We  would  rather  recommend  him 
to  omit  this  rule  until  he  is  acquainted  with  the  solu- 
tion of  equations  of  the  second  degree,  except  in  the 
cases  to  which  we  now  proceed.  In  arithmetic,  it 
must  be  observed  that  there  are  comparatively  very 
few  numbers  of  which  the  square  root  can  be  ex- 
tracted. For  example,  7  is  not  made  by  the  multipli- 
cation either  of  any  whole  number  or  fraction  by 
itself.  The  first  is  evident;  the  second  cannot  be 
readily  proved  to  the  beginner,  but  he  may,  by  taking 
a  number  of  instances,  satisfy  himself  of  this,  that  no 
fraction  which  is  really  such,  that  is  whose  numerator 
is  not  measured  by  its  denominator,  will  give  a  whole 
number  when  multiplied  by  itself,  thus  |  X  f  or  J^  is 
not  a  whole  number,  and  so  on.  The  number  7, 
therefore,  is  neither  the  square  of  a  whole  number,  nor 
of  a  fraction,  and,  properly  speaking,  has  no  square 
root.  Nevertheless,  fractions  can  be  found  extremely 
near  to  7,  which  have  square  roots,  and  this  degree 
of  nearness  may  be  carried  to  any  extent  we  please. 
Thus,  if  required,  between  7  and  7  ^^^^^  could 
be  found  a  fraction  which  has  a  square  root,  and  the 


EQUATIONS  OF  THE  SECOND  DEGREE.  13! 

fraction  in  the  last  might  be  decreased  to  any  extent 
whatever,  so  that  though  we  cannot  find  a  fraction 
whose  square  is  7,  we  may  nevertheless  find  one  whose 
square  is  as  near  to  7  as  we  please.  To  take  another 
example,  if  we  multiply  1-4142  by  itself  the  product 
is  1-99996164,  which  only  differs  from  2  by  the  very 
small  fraction  -00003836,  so  that  the  square  of  1-4142 
is  very  nearly  2,  and  fractions  might  be  found  whose 
squares  are  still  nearer  to  2.  Let  us  now  suppose  the 
following  problem.  A  man  buys  a  certain  number  of 
yards  of  stuff  for  two  shillings,  and  the  number  of 
yards  which  he  gets  is  exactly  the  number  of  shillings 

which  he  gives  for  a  yard.     How  many  yards  does  he 

2 
buy?     Let  x  be  this  number,  then  —  is  the  price  of 

2  x 

one  yard,  and  x=  —  or  #2  =  2.     This,  from  what  we 

have  said,  is  impossible,  that  is,  there  is  no  exact 
number  of  yards,  or  parts  of  yards,  which  will  satisfy 
the  conditions;  nevertheless,  1-4142  yards  will  nearly 
do  it,  1-4142136  still  more  nearly,  and  if  the  problem 
were  ever  proposed  in  practice,  there  would  be  no 
difficulty  in  solving  it  with  sufficient  nearness  for  any 
purpose.  A  problem,  therefore,  whose  solution  con- 
tains a  square  root  which  cannot  be  extracted,  may  be 
rendered  useful  by  approximation  to  the  square  root. 
Equations  of  the  second  degree,  commonly  called 
quadratic  equations,  are  those  in  which  there  is  the 
second  power,  or  square  of  an  unknown  quantity : 
such  as  x*  —  3  =  4*2  — 15,  .*2-f  3#  =  2*2  — x—  1,  etc. 


132  ON  THE  STUDY  OF  MATHEMATICS. 

By  transposition  of  their  terms,  they  may  always  be 
reduced  to  one  of  the  following  forms  : 


ax*  -\-bx-\-  <r  = 


For  example,  the  two  equations  given  above,  are 
equivalentto  3*2  —  12  =  0,  and  #2  —  4*  —  1—0,  which 
agree  in  form  with  the  second  and  last.  In  order  to 
proceed  to  each  of  these  equations,  first  take  the  equa- 
tion x*  =  a2.  This  equation  is  the  same  as  x2  —  a2  =  Q, 
or  (x-}-a)(x  —  0)  —  0.  Now,  in  order  that  the  pro- 
duct of  two  or  more  quantities  may  be  equal  to  noth- 
ing, it  is  sufficient  that  one  of  those  quantities  be  noth- 
ing, and  therefore  a  value  of  x  may  be  derived  from 
either  of  the  following  equations  : 

x—a  =  Q 
or  x-\-  a  =  Q 

the  first  of  which  gives  x  =  a,  and  the  second  x  =  —  a. 
To  elucidate  this,  find  x  from  the  following  equation  : 

(3  x  +  a)  (a*  -f  xs)  =  (**  -f  a  x)  (a2  +  a  x  -f  2  *2) 
develop  this  equation,  and  transpose  all  its  terms  on 
one  side,  when  it  becomes 


or    x  —  a 

or  O2  —  a*)(x*  —  2ax  —  «2)  =  0. 
This  last  equation  is  true  when  A2  —  «2  =  0,  or  when 


EQUATIONS  OF  THE  SECOND  DEGREE.  133 

x2  =  a2,  which  is  true  either  when  x=-\-a,  or  x  =  —  a. 
If  in  the  original  equation  -f  a  is  substituted  instead 
of  x,  the  result  is  4a  X  2as  =  2a2  X  4#2  ;  if  —  a  be 
substituted  instead  of  x,  the  result  is  0  =  0,  which 
show  that  -\-a  and  —  a  are  both  correct  values  of  x. 
We  have  here  noticed,  for  the  first  time,  an  equation 
of  condition  which  is  capable  of  being  solved  by  more 
than  one  value  of  x.  We  have  found  two,  and  shall 
find  more  when  we  can  solve  the  equation  x2  —  2  ax  — 
#2  =  0,  or  x2  —  2ax  =  a2.  Every  equation  of  the  sec- 
ond degree,  if  it  has  one  value  of  x,  has  a  second,  of 
which  x2  —  a2  is  an  instance,  where  x  =  ±a,  in  which 
by  the  double  sign  ±  is  meant,  that  either  of  them 
may  be  used  at  pleasure.  We  now  proceed  to  the  so- 
lution of  ax2  —  ^.#-f-<r—  0.  In  order  to  understand 
the  nature  of  this  equation,  let  us  suppose  that  we 
take  for  x  such  a  value,  that  ax2  —  bx-\-  c,  instead  of 
being  equal  to  0,  is  equal  to  j,  that  is 

y  =  axei  —  bx+c*  (1) 

in  which  the  value  of  y  depends  upon  the  value  given 
to  x,  and  changes  when  x  changes.  Let  m  be  one  of 
those  quantities  which,  when  substituted  instead  of  x, 
makes  ax2  —  bx  -\-  c  equal  to  nothing,  in  which  case  m 
is  called  a  root  of  the  equation, 

ax2  —  bx  +  c  =  §  (2) 

and  it  follows  that 

=  Q  (3) 


*In  the  investigations  which  follow,  a,  b,  and  c  are  considered  as  having 
the  sign  which  is  marked  before  them,  and  no  change  of  form  is  supposed  to 
take  place. 


134  ON  THE  STUDY  OF  MATHEMATICS. 

Subtract  (3)  from  (1),  the  result  of  which  is 


Here  y  is  evidently  equal  to  0,  when  x  =  m,  as  we 
might  expect  from  the  supposition  which  we  made  ; 
but  it  is  also  nothing  when  a(x-\-  m) — £  =  0;  there 
is,  therefore,  another  value  of  x,  for  which  j>:=0;  if 
we  call  this  n  we  find  it  from  the  equation  a  (n -\-rn~) — 
*=0, 

or  n-\-  m=  —  (4) 

In  (3)  substitute  for  b  its  value  derived  from  (4),  from 
which  b  =  a(n-\-m}\  it  then  becomes 

am1  —  a  m  (n  -f-  m~)  -\-c  =  0,  or  c — amn  —  Q, 

which  gives  m  n  =  — .  (5) 

Substitute  in  (1)  the  values  of  b  and  c  derived  from 
(4)  and  (5),  which  gives 

y  —  ax2  —  a(m-\-n)x-\-  amn 

=  a(x2 — m-\-n  x-\-mri). 

Now  the  second  factor  of  this  expression  arises  from 
multiplying  together  x  —  m  and  x — n  ;  therefore, 

y  =  a(x  —  m)  (x  —  »)  (6) 

To  take  an  example  of  this,  let  y  =  4x2  —  5#-fl. 
Here  when  x  =  l,  y  =  4:  —  5-fl  =  0,  and  therefore 
m  =  \.  If  we  divide  4 x2  —  5^-j-l  by  x — 1,  the  quo- 
tient (which  is  without  remainder)  is  4x  —  1,  and 

therefore 

y  =  (x— 1)(4*— 1). 

This  is  also  nothing  when  4x  — 1  =  0,  or  when  x  is  J. 
Therefore  n  =  £,  and  y  =  4  (x  —  1 )  (x  —  J) ,  a  result 


EQUATIONS  OF  THE  SECOND  DEGREE.  135 

coinciding  with  that  of  (6).  If,  therefore,  we  can  find 
one  of  the  values  of  x  which  satisfy  the  equation 
a  x2  —  b  x  -f-  c  =  0,  we  can  find  the  other  and  can  divide 
ax*  —  bx-\-c  into  the  factors  a,  x  —  m  and  x  —  n,  or 

ax2  —  bx-\-  c  =  a(x — m)  (x  —  «). 

If  we  multiply  x  -f-  m  by  x  -J-  #,  the  only  difference  be- 
tween (x  -j-  m)  (x  -f-  «)  and  (x  —  m)(x — «)  is  in  the 
sign  of  the  term  which  contains  the  first  power  of  x. 
If,  therefore, 

ax2  —  bx-{-  c  =  a(x  —  /«)  (x — «), 
it  follows  that 

ax2  -f-  b  x  -f-  c  =  a(x-\-m)  (x  -|-  ») . 
We  now  take  the  expression  <z  x2  —  b  x  —  c.   If  there 
is  one  value  of  x  which  will  make  this  quantity  equal 
to  0,  let  this  be  m,  and 

Let  y  =  ax2  —  bx  —  c 
Then  Q=am*  —  bm  —  c, 
from  which  y  =  a  (x2  —  ;/z2)  —  b  (x  —  m) 
=  (x — m)  (a  x-\-m  —  b) 
=  (x  —  m)  (a  x  -\-  a  m  —  ^) . 

Let be  called  «,  or  let  am  —  b  =  an  ',  then 

a 

y'=  (x  —  m)(ax-\-a  n) 
=  a(x  —  m)  (x -(-«). 

As  an  example,  it  may  be  shown  that 

3*2  — *  — 2  =  3(>  — l)O-ff). 

Again,  with  regard  to  ax2-\-bx  —  c,  since  (x-\-m) 
(x  —  n)  only  differs  from  (x  —  m)  (^  +  «)in  the  sign 


136  ON  THE  STUDY  OF  MATHEMATICS. 

of  the  term  which  contains  the  first  power  of  x,  it  is 
evident  that 

if  a  x2  —  b  x  —  c  =  a  (x — m)  (x  -f-  ri) 
a  x2  -j-  b  x  —  c  =  a  (x  -\-  m)  (x  —  n). 

Results  similar  to  those  of  the  first  case  may  be  ob- 
tained for  all  the  others,  and  these  results  may  be  ar- 
ranged in  the  following  way.  In  the  first  and  third, 
m  is  a  quantity,  which,  when  substituted  for  x,  makes 
y  =  0,  and  in  the  second  and  fourth  m  and  n  are  the 
same  as  in  the  first  and  third. 


1st.   .   .  y  =  ax2  —  bx-}-c  =  a(x  —  m)(x  —  ri) 

b  c 

m-\-n=  —       mn=  —  . 
a  a 


2d  .   .   .      = 


b  c 

m  4-  n  =  —       mn=  —  . 
a  a 


3d  .   .  .  y  =  ax2  —  bx  —  c  =  a(x  — 


b  c 

m  —  n—  —       mn=—. 
a  a 

4th    .   .    y  =  a  x2  -\-  b  x  —  c  —  a  (x  -\-  vi)  (x  —  n) 

b  c 

m  —  n  =  —       m  n  =  —  . 
a  a 

We  must  now  inquire  in  what  cases  a  value  can  be 
found  for  x}  which  will  make  jy  =  0  in  these  different 
expressions,  and  in  this  consists  the  solution  of  equa- 
tions of  the  second  degree. 

(1) 


EQUATIONS  OF  THE  SECOND  DEGREE.  137 


and  observe  that  (2ax  —  6)2  =  4a2x2  —  4abx-\-  P. 
Multiply  both  sides  of  (1)  by  4  a,  which  gives 

4ay  =  4a'>  x*  —  4abx  +  4ac  (2) 

Add  ft  to  the  first  two  terms  of  the  second  side  of  (2), 
and  subtract  it  from  the  third,  which  will  not  alter  the 
whole,  and  this  gives 


b*  (3) 

Now  it  must  be  recollected  that  the  square  of  any 
quantity  is  positive  whether  that  quantity  is  positive 
or  negative.  This  has  been  already  sufficiently  ex- 
plained in  saying  that  a  change  of  the  form  of  any  ex- 
pression does  not  change  the  form  of  its  square.  Com- 
mon multiplication  shows  that  (c  —  ^/)2  and  (d  —  r)2 
are  the  same  thing  ;  and,  since  one  of  these  must  be 
positive,  the  other  must  be  also  positive.  Whenever, 
therefore,  we  wish  to  say  that  a  quantity  is  positive,  it 
can  be  done  by  supposing  it  equal  to  the  square  of  an 
algebraical  quantity.  In  equation  (3)  there  are  three 
distinct  cases  to  be  considered. 

I.  When  ft2  is  greater  than  4ac,  that  is,  when 
b^  —  4ac  is  positive,  let  ft  —  4ac  —  &2,  which  expresses 
the  condition. 

Then  4ay  =  (2ax  —  £)2  —  &  (4) 

and  we  determine  those  values  of  x,  which  makejF=0, 
from  the  equation, 

(2ax  —  £)2  —  &  =  Q. 

We  have  already  solved  such  an  equation,  and  we 
find  that 


138  ON  THE  STUDY  OF  MATHEMATICS. 


where  either  sign  may  be  taken.     This  shows  that  y 
or  ax2  —  bx-\-  c  is  equal  to  nothing  either  when 


instead  of  x  is  put  -^  —  = 


^ 

Li  d  U  d 


the  second  values  being  formed  by  putting,  instead  of 


k  its  value  l£2  —  kac.     They  are  both  positive  quan- 

tities, because  &  being  equal  to  £2  —  ±ac  is  less  than 

b-\-  k 

ft,  and  therefore  k  is  less  than  b,  and  therefore  - 
^  _  ^  &a 

and  -     —  are  both  positive.     These  are  the  quantities 

«  & 

which  we  have  called  m  and  n  in  the  former  investiga- 

tions, and,  therefore, 

ax*  —  bx-\-c  =  a(x  —  m)(x  —  ri) 


=  a\x 


2a  )' 

Actual  multiplication  of  the  factors  will  show  that  this 
is  an  identical  equation. 

II.   When  /£2,  instead  of  being  greater  than  4tac,  is 
equal  to  it,  or  when  P  —  40<r  — 0  and  /&  =  0.     In  this 


case  the  values  of  m  and  n  are  equal,  each  being  -=— 

£a 

and 

/  b  \2 

y  =  ax^  —  bx-\-  c  =  a(x  —  m)(x  —  n)  =  a\x  —  ^—1  . 

In  this  case  y  is  said,   in  algebra,   to  be  a  perfect 
square,  since  its  square  root  can  be  extracted,  and  is 

V  a\x  —  —  j.     Arithmetically  speaking,   this  would 


EQUATIONS  OF  THE  SECOND  DEGREE.  139 

not  be  a  perfect  square  unless  a  was  a  number  whose 
square  root  could  be  extracted,  but  in  algebra  it  is 
usual  to  call  any  quantity  a  perfect  square  with  re- 
spect to  any  letter,  which,  when  reduced,  does  not 
contain  that  letter  under  the  sign  i/.  This  result  is 
one  which  often  occurs,  and  it  must  be  recollected 
that  when  P  —  lac  =  Q,  ax*  —  bx-\-c  is  a  perfect 
square. 

III.  When  b*  is  less  than  lac,  or  when  P  —  lac 
is  negative  and  lac  —  &2  positive,  let  lac  —  &  =  &, 
and  equation  (3)  becomes 


In  this  case  no  value  of  x  can  ever  make  y  =  0,  for 
the  equation  z;2  +  ^2  =  0  indicates  that  v2  is  equal  to 
a/2  with  a  contrary  sign,  which  cannot  be,  since  all 
squares  have  the  same  sign.  The  values  of  x  are  said, 
in  this  case,  to  be  impossible,  and  it  indicates  that 
there  is  something  absurd  or  contradictory  in  the  con- 
ditions of  a  problem  which  leads  to  such  a  result. 

Having  found  that  whenever 
ax1  — 


it  follows   that   a  x2  -f  b  x  -f  c  =  a  (x  -f-  m)  {x  -\-  «)  ,   we 
know  that 

(1)    when  b*  is  greater  than  lac, 


o  o 

la  2a 

(2)  when  fl  = 


140  ON  THE  STUDY  OF  MATHEMATICS. 

t>^ 


and  y  is  a  perfect  square  ; 

(3)  when  £2  is  less  than  4#r,  ax^  +  bx  +  c  cannot 
be  divided  into  factors. 

Now,  let 

y  =  ax*  —  bx—c  (1) 

As  before, 


(2) 

Let  P-\-±ac  =  &.     Then 

4ay  =  (2ax  —  £)2  —  P.  (3) 

Therefore  y  is  0  when  (2  ax  —  />)2  =  /£2,  or  when 


~ 


Now,  because  ft  is  less  than  ^2-j-4«^r,  b  is  less 
than  Vb*  -\-kac,  therefore  n  is  a  negative  quantity. 
Leaving,  for  the  present,  the  consideration  of  the 
negative  quantity,  we  may  decompose  (3)  into  factors 
by  means  of  the  general  formula 

/2_£2  =  (/_^)(/-f-^),  which  gives 
=  (2ax—-b  —  k}  (2ax  — 


\ 

from  which  y  or 


EQUATIONS  OF  THE  SECOND  DEGREE. 

ax*  —  bx  —  c  = 


V  P  +  ±ac  +b\(         V  P  +  ±a~c  —  b\ 


9 

Za 

Therefore,  from  what  has  been  proved  before, 
x  —  c  = 


The  following  are  some  examples,  of  the  truth  of 
which  the  student  should  satisfy  himself,  both  by  ref- 
erence to  the  ones  just  established,  and  by  actual 
multiplication  : 


1/49  — 24  \  f         7  —  1/49  — 

X ; 


o~\* 


X 


3-1/6 


1/221—9 


10      y  v  10 

If  we  collect  together  the  different  results  at  which 
we  have  arrived,  to  which  species  of  tabulation  the 
student  should  take  care  to  accustom  himself,  we  have 
the  following : 


*  Recollect  that  >7^  =  1/6X1  =  v  6~X  ^T=  2  ^67 


142  ON  THE  STUDY  OF  MATHEMATICS. 

ax*  —  bx-\-c  = 

a(X—b—VV9l'~^1'    }    {X—*""*    "»~~—    )     (B) 

\  / 

a 


—  bx  —  €•=• 


)  (D) 


These  four  cases  may  be  contained  in  one,  if  we 
apply  those  rules  for  the  change  of  signs  which  we 
have  already  established.  For  example,  the  first  side 
of  (C)  is  made  from  that  of  (A)  by  changing  the  sign 
of  c',  the  second  side  of  (C)  is  made  from  that  of  (A) 
in  the  same  way.  We  have  also  seen  the  necessity 
of  taking  into  account  the  negative  quantities  which 
satisfy  an  equation,  as  well  as  the  positive  ones ;  if 
we  take  these  into  account,  each  of  the  four  forms  of 
ax* -\-bx-\-c  can  be  made  equal  to  nothing  by  two 
values  of  x.  For  example,  in  (1),  when 


•*u  .  — 

either  #4  -  =  0 

Z  a 


L-  <7 

If  we  call  the  values  of  x  derived  from  the  equations 
m  and  n,  we  find  that 

1/>  —  ±ac  —b  —  VP  —  lac 


EQUATIONS  OF  THE  SECOND  DEGREE.  143 

In  the  cases  marked  (B),  (C),  and  (D),  the  results 


are 


— 

m= n= ^ (B 

2a  2a 


m= n= 5 (D) 

2a  2a 

and  in  all  the  four  cases  the  form  of  ax2-}- bx-\-c 
which  is  used,  is  the  same  as  the  corresponding  form 
of 

a(x — m~)  (x — ») 

and  the  following  results  may  be  easily  obtained.  In 
(A')  both  m  and  n,  if  they  exist  at  all,  are  negative. 
I  say,  if  they  exist  at  all,  because  it  has  been  shown 
that  if  b*  —  kac  is  negative,  the  quantity  ax2  -f  bx-\-  c 
cannot  be  divided  into  factors  at  all,  since  l/£2  —  ±ac 
is  then  no  algebraical  quantity,  either  positive  or  neg- 
ative. 

In  (B')  both,  if  they  exist  at  all,  are  positive. 

In  (C')  there  are  always  real  values  for  m  and  «, 
since  ^2-j-  kac  is  always  positive;  one  of  these  values 
is  positive,  and  the  other  negative,  and  the  negative 
one  is  numerically  the  greatest. 

In  (D')  there  are  also  real  values  of  m  and  «,  one 
positive  and  the  other  negative,  of  which  the  positive 
one  is  numerically  the  greatest.  Before  proceeding 
any  further,  we  must  notice  an  extension  of  a  phrase 


144  ON  THE  STUDY  OF  MATHEMATICS. 

which  is  usually  adopted.  The  words  greater  and  less, 
as  applied  to  numbers,  offer  no  difficult}'1,  and  from 
them  we  deduce,  that  if  a  be  greater  than  b,  a  —  c  is 
greater  than  b  —  c,  as  long  as  these  subtractions  are 
possible,  that  is,  as  long  as  c  can  be  taken  both  from 
a  and  b.  This  is  the  only  case  which  was  considered 
when  the  rule  was  made,  but  in  extending  the  mean- 
ing of  the  word  subtraction,  and  using  the  symbol 
—  3  to  stand  for  5  —  8,  the  principle  that  if  a  be  greater 
than  b,  a  —  c  is  greater  than  b  —  c,  leads  to  the  follow- 
ing results.  Since  6  is  greater  than  4,  6  — 12  is  greater 
than  4 — 12,  or — 6  is  greater  than — 8;  again  6  —  6 
is  greater  than  4  —  6,  or  0  is  greater  than  — 2.  These 
results,  particularly  the  last,  are  absurd,  as  has  been 
noticed,  if  we  continue  to  mean  by  the  terms  greater 
and  less,  nothing  more  than  is  usually  meant  by  them 
in  arithmetic  ;  but  in  extending  the  meaning  of  one 
term,  we  must  extend  the  meaning  of  all  which  are 
connected  with  it,  and  we  are  obliged  to  apply  the 
terms  greater  and  less  in  the  following  way.  Of  two 
algebraical  quantities  with  the  same  or  different  signs, 
that  one  is  the  greater  which,  when  both  are  connected 
with  a  number  numerically  greater  than  either  of  them, 
gives  the  greater  result.  Thus  —  6  is  said  to  be  greater 
than  —8,  because  20  —  6  is  greater  than  20  —  8,  0  is 
greater  than  — 4,  because  6  -f  0  is  greater  than  6  —  4  ; 
+  12  is  greater  than  — 30,  because  40 -f  12  is  greater 
than  40  —  30.  Nevertheless  — 30  is  said  to  be  numeri- 
cally greater  than  -f  12,  because  the  number  contained 


EQUATIONS  OF  THE  SECOND  DEGREE.  145 

in  the  first  is  greater  than  that  in  the  second.  For  this 
reason  it  was  said,  that  in  (C),  the  negative  quantity 
was  numerically  greater  than  the  positive,  because  any 
positive  quantity  is  in  algebra  called  greater  than  any 
negative  one,  even  though  the  number  contained  in 
the  first  should  be  less  than  that  in  the  second.  In 
the  same  way  — 14  is  said  to  lie  between  +  3  and 
—  20,  being  less  than  the  first  and  greater  than  the 
second.  The  advantage  of  these  extensions  is  the 
same  as  that  of  others  ;  the  disadvantage  attached  to 
them,  which  it  is  not  fair  to  disguise,  is  that,  if  used 
without  proper  caution,  they  lead  the  student  into 
erroneous  notions,  which  some  elementary  works,  far 
from  destroying,  confirm,  and  even  render  necessary, 
by  adopting  these  very  notions  as  definitions  ;  as  for 
example,  when  they  say  that  a  negative  quantity  is 
one  which  is  less  than  nothing  ;  as  if  there  could  be 
such  a  thing,  the  usual  meaning  of  the  word  less  being 
considered,  and  as  if  the  student  had  an  idea  of  a 
quantity  less  than  nothing  already  in  his  mind,  to 
which  it  was  only  necessary  to  give  a  name. 

The  product  (x  —  ni}  (x — »)  is  positive  when 
(x  —  m)  and  (x  —  n)  have  the  same,  and  negative  when 
they  have  different  signs.  This  last  can  never  happen 
except  when  x  lies  between  m  and  n,  that  is,  when  x 
is  algebraically  greater  than  the  one,  and  less  than 
the  other.  The  following  table  will  exhibit  this,  where 
different  products  are  taken  with  various  signs  of  m 
and  «,  and  three  values  are  given  to  x  one  after  the 


146  ON  THE  STUDY  OF  MATHEMATICS. 

other,  the  first  of  which  is  less  than  both  m  and  », 
the  second  between  both,  and  the  third  greater  than 
both. 

VALUE  OF  THE 

PRODUCT.  VALUE  OF  X.  PRODUCT  WITH 

ITS  SIGN. 


(x  — 

4)  (x  —  7) 

+  1 

+  18 

in  = 

+  4 

+  & 

—  2 

n  =  - 

-f-7 

+  10 

+  18 

(x  + 

10)<*—3) 

—  12 

+  30 

m  = 

-10 

—  7 

—  30 

n  = 

+  3 

+  4 

+  14 

(x  + 

2)O  +  12) 

-13 

+  11 

m  = 

—  2 

—  6 

—  24 

n== 

—  12 

—  1 

+  11 

The  student  will  see  the  reason  of  this,  and  per- 
form a  useful  exercise  in  making  two  or  three  tables 
of  this  description  for  himself.  The  result  is  that 
(# — m)(x  —  n)  is  negative  when  x  lies  between  m  and 
n,  is  nothing  when  x  is  either  equal  to  m  or  to  n,  and 
positive  when  x  is  greater  than  both,  or  less  than 
both.  Consequently,  a(x — m}(x  —  n)  has  the  same 
sign  as  a  when  x  is  greater  than  both  m  and  n,  or  less 
than  both,  and  a  different  sign  from  a  when  x  lies  be- 
tween both.  But  whatever  may  be  the  signs  of  a,  b, 
and  c,  if  there  are  two  quantities  m  arid  n,  which  make 

ax*  +  £*+  c  =  a(x  —  m)  (x  —  «), 

that  is,  if  the  equation  ax2  -\-  bx-\-  c  =  Q  has  real  roots, 
the  expression  ax2  +  bx  +  c  always  has  the  same  sign 


EQUATIONS  OF  THE  SECOND  DEGREE.  147 

as  a  for  all  values  of  x,  except  when  x  lies  between 
these  roots. 

It  only  remains  to  consider  those  cases  in  which 
ax2  -\-bx-\-c  cannot  be  decomposed  into  different  fac- 
tors, which  happens  whenever  b2 —  4ac  is  0,  or  nega- 
tive. In  the  first  case  when  b2  —  4ac  =  Q,  we  have 

^2 


b_ 

~2~a, 

and  as  these  expressions  are  composed  of  factors,  one 
of  which  is  a  square,  and  therefore  positive,  they  have 
always  the  same  sign  as  the  other  factor,  which  is  a. 
When  b2  —  4ac  is  negative,  we  have  proved  that  if 
y  =  a  x2  ±  b  x  -j-  *">  4ay  =  (2axd=.  £)2  -f-  k2,  where  k2  = 
kac  —  b2,  and  therefore  4  ay  being  the  sum  of  two 
squares  is  always  positive,  that  is,  ax2±bx-\-  c  has 
the  same  sign  as  a,  whatever  may  be  the  value  of  x. 
When  £  =  0,  the  expression  becomes  ax2  -\-frx,  or 
which  is  nothing  either  when  x  =  Q,  or 


when  ax-\-b-=§  and  x  =  --  ;  the  general  expres- 
sions for  m  and  n  become  in  this  case  -  5  --  and 
,  which  give  the  same  results. 


—   il 

When  ^  =  0,  the  expression  is  reduced  to  ax^-\- 
£  =  0,  which  is  nothing  when  x=  ±  •*/—  —  ,  which  is 
not  possible,  except  when  c  and  a  have  different  signs. 
In  this  case,  that  is,  when  the  expression  assumes  the 
form  ax2  —  c,  it  is  the  same  as 


148  ON  THE  STUDY  OF  MATHEMATICS. 

•Mi 

The  same  result  might  be  deduced  by  making  />  =  0 
in  the  general  expressions  for  m  and  n. 

When  0  —  0,  the  expression  is  reduced  to  bx-\-c, 
which  is  made  equal  to  nothing  by  one  value  of  x  only, 
that  is  -  -  —  .  If  we  take  the  general  expressions 
for  m  and  n,  and  make  a  —  Q  in  them,  that  is,  in 


,  and  -  —  ,  we  find  as 

Za  La 

the  results  -y-  and  —  ^  —  .  These  have  been  already  ex- 
plained. The  first  may  either  indicate  that  any  value 
of  x  will  solve  the  problem  which  produced  the  equa- 
tion ax"2  -f  bx  +  <:  =  0,  or  that  we  have  applied  a  rule 
to  a  case  which  was  not  contemplated  in  its  forma- 
tion, and  have  thereby  created  a  factor  in  the  numera- 
tor and  denominator  of  x,  which,  in  attempting  to 
apply  the  rule,  becomes  equal  to  nothing.  The  stu- 
dent is  referred  to  the  problem  of  the  two  couriers, 
solved  in  the  preceding  part  of  this  treatise.  The 
latter  is  evidently  the  case  here,  because  in  returning 
to  the  original  equation,  we  find  it  reduced  to  bx-\- 
<r  =  0,  which  gives  a  rational  value  for  xt  namely,  --  -. 

2  h 

The  second  value,  or  -  -^-,  which  in  algebraical  lan- 
guage is  called  infinite,  may  indicate,  that  though 
there  is  no  other  value  of  x,  except  -  -  —  ,  which 
solves  the  equation,  still  that  the  greater  the  number 
which  is  taken  for  x,  the  more  nearly  is  a  second  so- 


EQUATIONS  OF  THE  SECOND  DEGREE.        149 

lution  obtained.  The  use  of  these  expressions  is  to 
point  out  the  cases  in  which  there  is  anything  remark- 
able in  the  general  problem  ;  to  the  problem  itself  we 
must  resort  for  further  explanation. 

The  importance  of  the  investigations  connected 
with  the  expression  a  x2  -j-  b  x  -j-  c,  can  hardly  be  over- 
rated, at  least  to  those  students  who  pursue  mathe- 
matics to  any  extent.  In  the  higher  branches,  great 
familiarity  with  these  results  is  indispensable.  The 
student  is  therefore  recommended  not  to  proceed  until 
he  has  completely  mastered  the  details  here  given, 
which  have  been  hitherto  too  much  neglected  in  Eng- 
lish works  on  algebra. 

In  solving  equations  of  the  second  degree,  we  have 
obtained  a  new  species  of  result,  which  indicates  that 
the  problem  cannot  be  solved  at  all.  We  refer  to 
those  results  which  contain  the  square  root  of  a  nega- 
tive quantity.  We  find  that  by  multiplication  the 
squares  of  c  —  d  and  of  d — c  are  the  same,  both  being 
c*  —  2cd-\-  d*.  Now  either  c — d  or  d — c  is  positive, 
and  since  they  both  have  the  same  square,  it  appears 
that  the  squares  of  all  quantities,  whether  positive  or 
negative,  are  positive.  It  is  therefore  absurd  to  sup- 
pose that  there  is  any  quantity  which  x  can  represent, 
and  which  satisfies  the  equation  x*  =  —  a2,  since  that 
would  be  supposing  that  x2,  a  positive  quantity,  is 
equal  to  the  negative  quantity  — a2.  The  solution  is 
then  said  to  be  impossible,  and  it  will  be  easy  to  show 
an  instance  in  which  such  a  result  is  obtained,  and 


150  ON  THE  STUDY  OF  MATHEMATICS. 

also  to  show  that  it  arises  from  the  absurdity  of  the 
problem. 

Let  a  number  a  be  divided  into  any  two  parts,  one 
of  which  is  greater  than  the  half,  and  the  other  less. 

Call  the  first  of  these  -5-  -f  x,  then  the  second  must  be 
l. 

-^  —  x,  since  the  sum  of  both  parts  must  be  a.     Mul- 

a 

tiply  these  parts  together,  which  gives 


As  x  diminishes,  this  product  increases,  and  is  great- 
est of  all  when  x  =  Q,  that  is,  when  the  two  parts,  into 
which  a  is  divided,  are  -^  and  -^-,  or  when  the  number 

Z  L 

a  is  halved.     In  this  case  the  product  of  the  parts  is 

a        a         a2 

•o-  X  -K-,  or  —  ,  and  a  number  a  can  never  be  divided 

*        z  #2 

into  two  parts  whose  product  is  greater  than  -j.    This 

being  premised,  suppose  that  we  attempt  to  divide 
the  number  a  into  two  parts,  whose  product  is  b.    Let 
x  be  one  of  these  parts,  then  a  —  x  is  the  other,  and 
their  product  is  ax  —  x2. 
We  have,  therefore, 

ax  —  x*  =  b 
or  x2  —  ax-\-b  =  Q. 

If  we  solve  this  equation,  the  two  roots  are  the  two 
parts  required,  since  from  what  we  have  proved  of 
the  expression  x2  —  ax-}-  b  the  sum  of  the  roots  is  a 
and  their  product  b.  These  roots  are 


EQUATIONS  OF  THE  SECOND  DEGREE.  151 


which  are  impossible  when  -  --  b  is  negative,  or  when 

«2 
b  is  greater  than  ^-,  which  agrees  with  what  has  just 

been  proved,  that  no  number  is  capable  of  being  di- 

02 

vided  into  two  parts  whose  product  is  greater  than  -j. 
We  have  shown  the  symbol  V  —  a  to  be  void  of 
meaning,  or  rather  self-contradictory  and  absurd. 
Nevertheless,  by  means  of  such  symbols,  a  part  of 
algebra  is  established  which  is  of  great  utility.  It 
depends  upon  the  fact,  which  must  be  verified  by  ex- 
perience, that  the  common  rules  of  algebra  may  be 
applied  to  these  expressions  without  leading  to  any 
false  results.  An  appeal  to  experience  of  this  nature 
appears  to  be  contrary  to  the  first  principles  laid  down 
at  the  beginning  of  this  work.  We  cannot  deny  that 
it  is  so  in  reality,  but  it  must  be  recollected  that  this 
is  but  a  small  and  isolated  part  of  an  immense  sub- 
ject, to  all  other  branches  of  which  these  principles 
apply  in  their  fullest  extent.  There  have  not  been 
wanting  some  to  assert  that  these  symbols  may  be 
used  as  rationally  as  any  others,  and  that  the  results 
derived  from  them  are  as  conclusive  as  any  reasoning 
could  make  them.  I  leave  the  student  to  discuss  this 
question  as  soon  as  he  has  acquired  sufficient  knowl- 
edge to  understand  the  various  arguments:  at  present 

*The  general  expressions  for  in  and  «  give  a  ~  —  a      *     as  the  roots  of 


—  ax 


152  ON  THE  STUDY  OF  MATHEMATICS. 

let  him  proceed  with  the  subject  as  a  part  of  the 
mechanism  of  algebra,  on  the  assurance  that  by  care- 
ful attention  to  the  rules  laid  down  he  can  never  be 
led  to  any  incorrect  result.  The  simple  rule  is,  apply 
all  those  rules  to  such  expressions  as  V  —  a,  a  -f  V  —  b, 
etc.,  which  have  been  proved  to  hold  good  for  such 
quantities  as  \/a,  a  -f-  1/^,  etc.  Such  expressions  as 
the  first  of  these  are  called  imaginary,  to  distinguish 
them  from  the  second,  which  are  called  real;  and  it 
must  always  be  recollected  that  there  is  no  quantity, 
either  positive  or  negative,  which  an  imaginary  ex- 
pression can  represent. 

It  is  usual  to  write  such  symbols  as  V  —  b  in  a  dif- 
ferent form.  To  the  equation  —  £  =  ^x(  —  1)  apply 
the  rule  derived  from  the  equation  V  '  xy  =  V  'xX  V  yt 
which  gives  V  —  b  =  Vb  X  V-  —  1,  of  which  the  first 
factor  is  real  and  the  second  imaginary.  Let  J/£=*i 
then  I/  —  b  =  cV/-—  1.  In  this  way  all  expressions 
may  be  so  arranged  that  j/  —  1  shall  be  the  only  im- 
aginary quantity  which  appears  in  them.  Of  this  re- 
duction the  following  are  examples  : 

V  '-^24  =  1/24  1/—  1  =  2  1/6V—1 


The  following  tables  exhibit  other  applications  of 
the  rules  : 


EQUATIONS  OF  THE  SECOND  DEGREE.  153 


c  =aV--l  ci^  —  aty 

^•2  — —  a2  cs  =  as,  etc. 


=  —  a 


The  powers  of  such  an  expression  as  a\/  —  1  are 
therefore  alternately  real  and  imaginary,  and  are  posi- 
tive and  negative  in  pairs. 


ac— 


Let  the  roots  of  the  equation  ax*  -\-  frx-\-c  =  Q  be  im- 
possible, that  is,  let  £2  —  ±ac  be  negative  and  equal 
to  —  £2.  Its  roots,  as  derived  from  the  rules  estab- 
lished when  £2  —  ±ac  was  positive,  are 


__ 
and  -  •=  --  ,  or 


2a 


Take  either  of  these  instead  of  x  ;   for  example,  let 
b          k       — 


154  ON  THE  STUDY  OF  MATHEMATICS. 

P  bk    /— T  £2 

Then  a  x2  =  -.  -  V — 1  —   . 

4a        2a  40 

ffi         bk 


-35  +  H"- 


Therefore,     ax"*-}-bx-\-c=--  —  -  —  \-  c,    in 

4a          4a         &a 

which,  if  ±ac  —  &  be  substituted  instead  of  /&2,  the  re- 
sult is  0.  It  appears,  then,  that  the  imaginary  expres- 
sions which  take  the  place  of  the  roots  when  ft  —  4ac 
is  negative,  will,  if  the  ordinary  rules  be  applied,  pro- 
duce the  same  results  as  the  roots.  They  are  thence 
called  imaginary  roots,  and  we  say  that  every  equa- 
tion of  the  second  degree  has  two  roots,  either  both 
real  or  both  imaginary.  It  is  generally  true,  that 
wherever  an  imaginary  expression  occurs  the  same  re- 
sults will  follow  from  the  application  of  these  expres- 
sions in  any  process  as  would  have  followed  had  the 
proposed  problem  been  possible  and  its  solution  real. 
When  an  equation  arises  in  which  imaginary  and 
real  expressions  occur  together,  such  as  a  -f  b  V  -  -  1  = 
c  +  dV  —  1,  when  all  the  terms  are  transferred  on  one 
side,  the  part  which  is  real  and  that  which  is  imagin- 
ary must  each  of  them  be  equal  to  nothing.  The 
equation  just  given  when  its  left  side  is  transposed 
becomes  a  —  c-\-(b  —  </)!/  —  1=0.  Now,  if  b  is  not 
equal  to  dy  let  b  —  d=e;  then  a  —  c  -\-  eV  —  1  =  0,  and 

/  —  T      c  —  a 
V  —  1  =  --  ;    that   is,   an   imaginary  expression  is 

equal  to  a  real  one,  which  is  absurd.  Therefore,  b  =  d 


EQUATIONS  OF  THE  SECOND  DEGREE.  155 

and  the  original  equation  is  thereby  reduced  to  a  =  c. 
This  goes  on  the  supposition  that  a,  b,  c,  and  d  are 
real.  If  they  are  not  so  there  is  no  necessary  absurd- 
ity in  V  —  1  =  —  —  .  If,  then,  we  wish  to  express 
that  two  possible  quantities  a  and  b  are  respectively 
equal  to  two  others  c  and  d,  it  may  be  done  at  once  by 
the  equation 


The  imaginary  expression  I/  —  a  and  the  negative  ex- 
pression •  —  b  have  this  resemblance,  that  either  of 
them  occurring  as  the  solution  of  a  problem  indicates 
some  inconsistency  or  absurdity.  As  far  as  real  mean- 
ing is  concerned,  both  are  equally  imaginary,  since 
0  —  a  is  as  inconceivable  as  V  —  a.  What,  then,  is  the 
difference  of  signification?  The  following  problems 
will  elucidate  this.  A  father  is  fifty-six,  and  his  son 
twenty-nine  years  old  :  when  will  the  father  be  twice 
as  old  as  the  son?  Let  this  happen  x  years  from  the 
present  time  ;  then  the  age  of  the  father  will  be  56  -|-  x, 
and  that  of  the  son  29  -f#;  and  therefore,  56-f-#  = 
2  (29  -f  #)  =  58  -j-  2  x,  or  x  =  —  2.  The  result  is  ab- 
surd ;  nevertheless,  if  in  the  equation  we  change  the 
sign  of  x  throughout  it  becomes  56  —  #  =  58  —  2x,  or 
x  =  2.  This  equation  is  the  one  belonging  to  the 
problem  :  a  father  is  56  and  his  son  29  years  old  ; 
when  was  the  father  twice  as  old  as  the  son  ?  the  an- 
swer to  which  is,  two  years  ago.  In  this  case  the 
negative  sign  arises  from  too  great  a  limitation  in  the 


156  ON  THE  STUDY  OF  MATHEMATICS. 

terms  o!  the  problem,  which  should  have  demanded 
how  many  years  have  elapsed  or  will  elapse  before  the 
father  is  twice  as  old  as  his  son  ? 

Again,  suppose  the  problem  had  been  given  in  this 
last-mentioned  way.  In  order  to  form  an  equation,  it 
will  be  necessary  either  to  suppose  the  event  past  or 
future.  If  of  the  two  suppositions  we  choose  the 
wrong  one,  this  error  will  be  pointed  out  by  the  nega- 
tive form  of  the  result.  In  this  case  the  negative  re- 
sult will  arise  from  a  mistake  in  reducing  the  problem 
to  an  equation.  In  either  case,  however,  the  result 
may  be  interpreted,  and  a  rational  answer  to  the  ques- 
tion may  be  given.  This,  however,  is  not  the  case  in 
a  problem,  the  result  of  which  is  imaginary.  Take 
the  instance  above  solved,  in  which  it  is  required  to 
divide  a  into  two  parts,  whose  product  is  b.  The  re- 
sulting equation  is 


a 

or  x=  -=- 


the  roots  of  which  are  imaginary  when  b  is  greater 
than  ^-.  If  we  change  the  sign  of  x  in  the  equation 
it  becomes 


* 

'  ~ 


and  the  roots  of  the  second  are  imaginary,  if  those  of 
the  first  are  so.  There  is,  then,  this  distinct  difference 


EQUATIONS  OF  THE  SECOND  DEGREE.  157 

between  the  negative  and  the  imaginary  result.  When 
the  answer  to  a  problem  is  negative,  by  changing  the 
sign  of  x  in  the  equation  which  produced  that  result, 
we  may  either  discover  an  error  in  the  method  of 
forming  that  equation  or  show  that  the  question  of  the 
problem  is  too  limited,  and  may  be  extended  so  as  to 
admit  of  a  satisfactory  answer.  When  the  answer  to 
a  problem  is  imaginary  this  is  not  the  case. 


CHAPTER  XI. 

ON  ROOTS  IN  GENERAL,  AND  LOGARITHMS. 


'T^HE  meaning  of  the  terms  square  root,  cube  root, 
-*-  fourth  root,  etc.,  has  already  been  defined.  We 
now  proceed  to  the  difficulties  attending  the  connex- 
ion of  the  roots  of  a  with  the  powers  of  a.  The  fol- 
lowing table  will  refresh  the  memory  of  the  student 
with  respect  to  the  meaning  of  the  terms  : 

NAM  EOF  JT.  NAME  OF  JT. 

Square  of  a  ----  -  x=aa  Square  Root  of  a  -  -         xx=a 

Cube  of  a  ------  x=aaa  Cube  Root  of  a  -  -  -       xxx=a 

Fourth  Power  of  a  -  x=aaaa  Fourth  Root  of  a  -  -     xxxx=a 

Fifth  Power  of  a  -  -  x=aaaaa  Fifth  Root  of  a  -  -  -  xxxxx=a 

The  different  powers  and  roots  of  a  have  hitherto 
been  expressed  in  the  following  way  : 

Powers  a2  #3  #4  a5  .  .  am  .  .  am+n,  etc. 
Roots   f/^*    ^a    Va    V^   Va    m+t/^,  etc. 

which  series  are  connected  together  by  the  following 
equation,  (v/a)n  =  a. 

*The  z  is  usually  omitted,  and  the  square  root  is  written  thus  iTa. 


ROOTS  IN  GENERAL  AND  LOGARITHMS.       I5Q 

There  has  hitherto  been  no  connexion  between  the 
manner  of  expressing  powers  and  roots,  and  we  have 
found  no  properties  which  are  common  both  to  powers 
and  roots.  Nevertheless,  by  the  extension  of  rules, 
we  shall  be  led  to  a  method  of  denoting  the  raising  of 
powers,  the  extraction  of  roots,  and  combinations  of 
the  two,  to  which  algebra  has  been  most  peculiarly 
indebted,  and  the  importance  of  which  will  justify  the 
length  at  which  it  will  be  treated  here. 

Suppose  it  required  to  find  the  cube  of  202^3;  that 
is,  to  find  2  a?  ^  X  2  a2  P  X  2  a2  &.  The  common  rules 
of  multiplication  give,  as  the  result,  806^9,  which  is 
expressed  in  the  following  equation, 


Similarly 


2    a]        64    aQ  ' 
and  the  general  rule  by  which  any  single  term  may  be 
raised  to  the  power  whose  index  is  n,  is  :  Raise  the  co- 
efficient to  the  power  «,   and  multiply  the  index  of 
every  letter  by  n,  that  is, 

(a?  bq  cry  —  anp  bnq  cnr. 

In  extracting  the  root  of  any  simple  term,  we  are 
guided  by  the  manner  in  which  the  corresponding 
power  is  found.  The  rule  is  :  Extract  the  required  root 
of  the  coefficient,  and  divide  the  index  of  each  letter 
by  the  index  of  the  root.  Where  these  divisions  do 
not  give  whole  numbers  as  the  quotients,  the  expres- 


160  ON  THE  STUDY  OF  MATHEMATICS. 

sion  whose  root  is  to  be  extracted  does  not  admit  of 
the  extraction  without  the  introduction  of  some  new 
symbol.  For  example,  extract  the  fourth  root  of 
16a12£M,  or  find  f/16012^M.  The  expression  here 
given  is  the  same  as  the  following  : 


or  (2#3<£2^)4,  the  fourth  root  of  which  is  203^2^,  con- 
formably to  the  rule. 

Any  root  of  a  product,  such  as  AB,  may  be  ex- 
tracted by  extracting  the  root  of  each  of  its  factors. 
Thus,  T^ZZ?  =  ^  ~A  f£.  For,  raise  ^A  V~B  to  the 
third  power,  the  result  of  which  is, 


x 

or  iTZ  f/A-  V~A  x  f^ 

or  AB. 

In  the  same  way  it  may  be  proved  generally,  that 
VABC=vrAvrBV^  The  most  simple  way  of  rep- 
resenting any  root  of  any  expression  is  the  dividing  it 
into  two  factors,  one  of  which  is  the  highest  which  it 
admits  of  whose  root  can  be  extracted  by  the  rule  just 


given.  For  example,  in  finding  fl6  a^b1  c  we  must 
observe  that  16  is  8  X  2,  at  is  as  X  ^,  b1  is  b*  X  ^  and 
the  expression  is  8  a*  b*  X  2  a  b  c,  the  cube  root  of  which, 
found  by  extracting  the  cube  root  of  each  factor,  is 
The  second  factor  has  no  cube  root 


which   can   be  expressed  by  means  of  the  symbols 
hitherto  used,  but  when  the  numbers  which  a,  b,  and 


c  stand  for  are  known,  'Zabc  maybe  found  either 


ROOTS  IN  GENERAL  AND  LOGARITHMS.  l6l 

exactly,  or,  when  that  is  not  possible,  by  approxima- 
tion. 

We  find  that  a  power  of  a  power  is  found  by  affix- 
ing, as  an  index,  the  product  of  the  indices  of  the  two 
powers.  Thus  O2)4  or  a2  X  <$  X  #2  X  «2  is  a8,  or  04X2. 
This  is  the  same  as  (#4)2,  which  is  cfi  X  #4>  or  as. 
Therefore,  generally  (am)n  =  (aM)m  =  amn.  In  the  same 
manner,  a  root  of  a  root  is  the  root  whose  index  is  the 
product  of  the  indices  of  the  two  roots.  Thus 


For  since  a  =  %/a  Va  l/a  X  V~a  V~a  i/a,  the  square 
root  of  a  is  I/a  i/a  v/fl,  the  cube  root  of  which  is  Va. 
This  is  the  same  as  v  i/a,  and  generally 

m  /  n  /  n  /  nt  /^  mn  / 

V  y  a  =  v    v  a  =    v  a. 

Again,  when  a  power  is  raised  and  a  root  extracted, 
it  is  indifferent  which  is  done  first.  Thus  f7^2  is  the 
same  thing  as  (^  a  )  2.  For  since  a2  =  a^a,  the  cube 
root  may  be  found  by  taking  the  cube  root  of  each  of 
these  factors,  that  is  it/a?  =  f/aX  f/^=(f/a)2)  and 
generally 


In  the  expression  v7^,  n  and  m  may  both  be  mul- 
tiplied by  any  number,  without  altering  the  expres- 
sion, that  is 


To  prove  this,  recollect  that 


I  62  ON  THE  STUDY  OF  MATHEMATICS. 

But  a"1*  is  (a1*)',  and  by  definition  \/(amy  =  am.  There- 
fore "\/am*  =  \/~cr.  This  multiplication  is  equivalent 
to  raising  a  power  of  1/0"*,  and  afterwards  reducing 

the  result  to  its  former  value,  by  extracting  the  corre- 

mp    .  m 

spending  root,  in  the  same  way  as  — —  signifies  that  - 

np  n 

has  been  multiplied  by/,  and  the  result  has  been  re- 
stored to  its  former  value  by  dividing  it  by  /. 

The  following  equations  should  be  established  by 
the  student  to  familiarise  him  with  the  notation  aid 
principles  hitherto  laid  down. 


« — »z/7 7T — j 

1  -f-  b}n~m  X      V^(«  —  ^)"+WI  = 
(cP  —  P) 


n\ab  __  v^ab  __  {/a  yHJ  _     n\a_         *\b 

\^  ="  V^  ="  'VTVd  =  :  \^  '  '  W 


The  quantity  \/am  is  a  simple  expression  when  m 
can  be  divided  by  n,  without  remainder,  for  example 
i/a12  =  a*,  f/020  =  a4,  and  in  general,  whenever  m  can 

•m 

be  divided  by  n  without  remainder,  i/7^"1  =  an  .  This 
symbol,  viz.,  a  letter  which  has  an  exponent  appear- 
ing in  a  fractional  form,  has  not  hitherto  been  used. 
We  may  give  it  any  meaning  which  we  please,  pro- 

vided it  be  such  that  when  —  is  fractional  in  form  only, 

n 

and  not  in  reality,   that  is,  when  m  is  divisible  by 


ROOTS  IN  GENERAL  AND  LOGARITHMS.       163 

— 

n,   and  the   quotient  is  /,   an  shall  stand  for  ap,   or 

•m 

aaa (/)*•    It  will  be  convenient  to  let  an  always 

stand  for  i/0™,  in  which  case  the  condition  alluded  to 

WL  — 

is  fulfilled,  since  when  —  — /,  cF  or  {X5"  =  a*.     This 
n 

extension  of  a  rule,  the  advantages  of  which  will  soon 
be  apparent,  is  exemplified  in  the  following  table, 
which  will  familiarise  the  student  with  the  different 
cases  of  this  new  notation  : 

a%  stands  for  i/a1  or  I/a 

a%  stands  for  ^ a 

a*  stands  for  \/a 
a%  stands  for  f7^  or  (f/^  )  2 
al  stands  for  I/a*  or  (Va  ) 7 


(P  +  ^)   2    stands  for  V  (p  +  q} 

stands  for 

(-i\\     . 
d»y  stands  for 

The  results  at  which  we  have  arrived  in  this  chap- 
ter, translated  into  this  new  language,  are  as  follows  : 

^')  "=(*")"=*  (!) 

BC)"  =  A"  £"  C"  (2) 

*This  is  a  notation  in  common  use,  and  means  that  aaa is  to  be 

continued  until  it  has  been  repeated/  times.    Thus 

a+a  +  a+ (/)=/«, 

a  X  a  X  a  X (?!  =  <**. 


164  ON  THE  STUDY  OF  MATHEMATICS. 

f    l\-  1 

(a*  )*,=  *«  (3) 

/         \1  /     i\*»  m 

{am)n  =  \an)    =an  (4) 

«"  =  £  (5) 

The  advantages  resulting  from  the  adoption  of  this 
notation,  are,  (1)  that  time  is  saved  in  writing  alge- 
braical expressions  ;  (2)  all  rules  which  have  been 
shown  to  hold  good  for  performing  operations  upon 
such  quantities  as  am,  hold  good  also  for  performing 
the  same  operations  upon  such  quantities  as  a",  in 
which  the  exponents  are  fractional.  The  truth  of  this 
last  assertion  we  proceed  to  establish. 

Suppose  it  required  to  multiply  together  a"  and 
± 
an,  or  Va™  and  Val.     From  (2)  this  is  V  a"1  X  *',  or 

1/am+i,  or  a"*".     Suppose  it  now  required  to  multiply 

w  p_ 

an   and  aq  '.     From  (5)  the  first  of  these  is  the  same  as 

*n?  ftf 

anq  ,  and  the  second  is  the  same  as  anq.     The  product 

mq-\-np 

of  these  by  the  last  case  is  a   n«  ,  or  T/0l**+"->.     But 

m  g  -|-  np  .     m         p 

—  —  -  —  -  is  ---  h  —  »  an(i  therefore 


nq 

— 


an  X  <*q  =a 


/c\ 


This  is  the  same  result  as  was  obtained  when  the 
indices  were  whole  numbers.  The  rule  is  :  To  multi- 
ply together  two  powers  of  the  same  quantity,  add 
the  indices,  and  make  the  sum  the  index  of  the  pro- 
duct. It  follows  in  the  same  way  that 


ROOTS  IN  GENERAL  AND  LOGARITHMS.  165 


or,  to  divide  one  power  of  a  quantity  by  another,  sub- 
tract the  index  of  the  divisor  from  that  of  the  divi- 
dend, and  make  the  difference  the  index  of  the  result. 

Suppose  it  required  to  find  \a")  .     It  is  evident 

tn_  tn  ™i™  zm  f     fn\  ^  2"z 

that  an  X  an  =  a"  *  =  «""",  or  \a* )    =  a*~.    Similarly 

/     ™\3  yn  /     ™y»  rnp 

\a*J    =  an  ,    and    so    on.     Therefore    \an )    —  a n , 
Again  to  find  (/**)*,  or  V  a" .     Let  this  be  a? . 

Then  a*  =  V  a" ,  or  (a*)  =  a",  or  a~  -.=  a".   There- 

xq        m          x        m  (  ^\j         ZL 

fore  -  -  =  — ,  or  —  =  — ,  and  \an  )    =  anq • 
y          n          y        nq 

(  ™\2-          */~7  -V 
Again  to  find  \an  )    or  ]/     \an )  .     Apply  the  last 

/    ™y  m£ 

two   rules,    and   it   appears   that    \an )    =  an  ,    and 

g  /    mp  mp  /    ™\2-  mp  ^X  — 

V  an  =  a"*  .     Therefore  {a* )q  =  an*  =  an    * . 

The  rule  is :  To  raise  one  power  of  a  quantity  to 
another  power,  multiply  the  indices  of  the  two  powers 
together,  and  make  the  product  the  index  of  the  re- 
sult. All  these  rules  are  exactly  those  which  have 
been  shown  to  hold  good  when  the  indices  are  whole 
numbers.  But  there  still  remains  one  remarkable  ex- 
tension, which  will  complete  this  subject. 

We  have  proved  that  whether  m  and  n  be  whole 


a 


or  fractional   numbers,  —  =  am~n.     The   only   cases 

,,«  * 


a 


which  have  been  considered  in  forming  this  rule  are 


1 66  ON  THE  STUDY  OF  MATHEMATICS. 

those  in  which  m  is  greater  than  #,  being  the  only 
ones  in  which  the  subtraction  indicated  is  possible.  If 
we  apply  the  rule  to  any  other  case,  a  new  symbol  is 

produced,  which  we  proceed  to  consider.     For  exam- 

a? 
pie,  suppose  it  required  to  find  — .     If  we  apply  the 

rule,  we  find  the  result  #3~7,  or  a-*,  for  which  we  have 
hitherto  no  meaning.  As  in  former  cases,  we  must 
apply  other  methods  to  the  solution  of  this  case,  and 
when  we  have  obtained  a  rational  result,  cr*  may  be 

used  in  future  to  stand  for  this  result.  Now  the  frac- 
as3 1 

tion  -=  is  the  same  as  -7,  which  is  obtained  by  divid- 
a1  a* 

ing  both  its  numerator  and  denominator  by  aB.  There- 
fore —r  is  the  rational  result,  for  which  we  have  ob- 
a* 

tained  a"4  by  applying  a  rule  in  too  extensive  a  manner. 

Nevertheless,   if   or*   be  made  to  stand  for  -r,    and 

1  a 

a-™  for  — ,  the  rule  will  always  give  correct  results, 

and  the  general  rules  for  multiplication,  division,  and 
raising  of  powers  remain  the  same  as  before.  For 
example,  *—x<r~  is  —  X  — ,  or  -^,  which  is 
—^,  or  tf-^+'O,  or  arm-H.  Similarly 


a—nt  ant  an 

IF*'  or  T'  is  ^'  or  an~m>  or  *""    ~n) 


Again 

(^)"Wis^,or-L,  orir-, 

and  so  on. 


ROOTS  IN  GENERAL  AND  LOGARITHMS.      167 

It  has  before  been  shown  that  a°  stands  for  1  when- 
ever it  occurs  in  the  solution  of  a  problem.  We  can 
now,  therefore,  assign  a  meaning  to  the  expression  a™, 
whether  m  be  whole  or  fractional,  positive,  negative, 
or  nothing,  and  in  all  these  cases  the  following  rules 
hold  good  : 


a™ 

—  =  am~"  =  a"1  a 
an 


The  student  can  now  understand  the  meaning  of  such 
an  expression  as  10  -301,  where  the  index  or  exponent 
is  a  decimal  fraction.  Since  .301  is  •£$}$,  this  stands 
for  lcxj//(10)3°1,  an  expression  of  which  it  would  be  im- 
possible to  calculate  the  value  by  any  method  which 
the  student  has  hitherto  been  taught,  but  which  may 
be  shown  by  other  processes  to  be  very  nearly  equal 
to  2. 

Before  proceeding  to  the  practice  of  logarithmic 
calculations,  the  student  should  thoroughly  under- 
stand the  meaning  of  fractional  and  negative  indices, 
and  be  familiar  with  the  operations  performed  by 
means  of  them.  He  should  work  many  examples  of 
multiplication  and  division  in  which  they  occur,  for 
which  he  can  have  recourse  to  any  elementary  work. 
The  rules  are  the  same  as  those  to  which  he  has  been 
accustomed,  substituting  the  addition,  subtraction, 
and  so  forth,  of  fractional  indices,  instead  of  these 
which  are  whole  numbers. 


l68  ON  THE  STUDY  OF  MATHEMATICS. 

In  order  to  make  use  of  logarithms,  he  must  pro- 
vide himself  with  a  table.  Either  of  the  following 
works  may  be  recommended  to  him  : 

[1.  Bruhns,  A  New  Manual  of  Logarithms  to  Seven 
Places  of  Decimals  (English  preface,  Leipsic). 

2.  Schron,  Seven-Figure  Logarithms  (English  edi- 
tion, London). 

3.  Bremiker's  various  editions  of  Vega's  Logarith- 
mic Tables  (Weidmann,  Berlin).     With  English  pref- 
ace.] 

4.  Callet,  Tables  portatives  de  Logarithmes.     (Last 
impression,  Paris,  1890). 

5.  V.  Caillet,    Tables  des  Logarithmes  et   Co-Loga- 
rithmes  des  nombres  (Paris). 

6.  Hutton's  Mathematical  Tables  (London). 

7.  Chambers's  Mathematical  Tables  (Edinburgh). 

8.  The  American  six-figure  Tables  of   Jones,  of 
Wells,  and  of  Haskell. 

For  fuller  bibliographical  information  on  the  sub- 
ject of  tables  of  logarithms,  see  the  Encyclopedia  Bri- 
tannica,  Article  "Tables,"  Vol.  XXIIL—  Ed.]* 

The  limits  of  this  treatise  will  not  allow  us  to  enter 


*The  original  text  of  De  Morgan,  for  which  the  above  paragraph  has 
been  substituted,  reads  as  follows  :  "  Either  of  the  following  works  may  be 
recommended  to  him:  (i)  Taylor's  Logarithms.  (2)  Hutton's  Logarithms. 
(3)  Babbage,  Logarithms  of  Numbers;  Callet,  Logarithms  of  Sines,  Cosines, 
etc.  (4)  Bagay,  Tables  Astronomiques  et  Hydrographiques.  The  first  and  last 
of  these  are  large  works,  calculated  for  the  most  accurate  operations  of 
spherical  trigonometry  and  astronomy.  The  second  and  third  are  better 
suited  to  the  ordinary  student.  For  those  who  require  a  pocket  volume  there 
are  Lalande's  and  Hassler's  Tables,  the  first  published  in  France,  the  second 
in  the  United  States."— Ed. 


ROOTS  IN  GENERAL  AND  LOGARITHMS.       169 

into  the  subject  of  the  definition,  theory,  and  use  of 
logarithms,  which  will  be  found  fully  treated  in  the 
standard  text-books  of  Arithmetic,  Algebra,  and  Trig- 
onometry. There  is,  however,  one  consideration  con- 
nected with  the  tables,  which,  as  it  involves  a  princi- 
ple of  frequent  application,  it  will  be  well  to  explain 
here.  On  looking  into  any  table  of  logarithms  it  will 
be  seen,  that  for  a  series  of  numbers  the  logarithms 
increase  in  arithmetical  progression,  as  far  as  the  first 
seven  places  of  decimals  are  concerned  ;  that  is,  the 
difference  between  the  successive  logarithms  continues 
the  same.  For  example,  the  following  is  found  from 
any  tables  : 

Log.  41713  =  4.6202714 

Log.  41714  =  4.6202818 

Log.  41715  =  4.6202922 

The  difference  of  these  successive  logarithms  and  of 
almost  all  others  in  the  same  page  is  .0000104.  There- 
fore in  this  the  addition  of  1  to  the  number  gives  an 
addition  of  .0000104  to  the  logarithm.  It  is  a  general 
rule  that  when  one  quantity  depends  for  its  value  upon 
another,  as  a  logarithm  does  upon  its  number,  or  an 
algebraical  expression,  such  as  x2  -\-  x  upon  the  letter 
or  letters  which  it  contains,  if  a  very  small  addition  be 
made  to  the  value  of  one  of  these  letters,  in  conse- 
quence of  which  the  expression  itself  is  increased  or 
diminished;  generally  speaking,  the  increment*  of  the 

*  When  any  quantity  is  increased,  the  quantity  by  which  it  is  increased  is 
called  its  increment. 


170  ON  THE  STUDY  OF  MATHEMATICS. 

expression  will  be  very  nearly  proportional  to  the  in- 
crement of  the  letter  whose  value  is  increased,  and  the 
more  nearly  so  the  smaller  is  the  increment  of  the  let- 
ter. We  proceed  to  illustrate  this.  The  product  of 
two  fractions,  each  of  which  is  less  than  unity,  is  itself 
less  than  either  of  its  factors.  Therefore  the  square, 
cube,  etc.,  of  a  fraction  less  than  unity  decrease,  and 
the  smaller  the  fraction  is  the  more  rapid  is  that  de- 
crease, as  the  following  examples  will  show  : 

Let  x  =.01  Let  x  =.00001 

x2 = . 0001  x2 = . 0000000001 

x* = . 000001  xs = . 000000000000001 

etc.  etc. 

Now  quantities  are  compared,  not  by  the  actual 
difference  which  exists  between  them,  but  by  the  num- 
ber of  times  which  one  contains  the  other,  and,  of  two 
quantities  which  are  both  very  small,  one  may  be  very 
great  as  compared  with  the  other.  In  the  second  ex- 
ample x2  and  XB  are  both  small  fractions  whem  com- 
pared with  unity  ;  nevertheless,  x2  is  very  great  when 
compared  with  xs,  being  100,000  times  its  magnitude. 
This  use  of  the  words  small  and  great  sometimes  em- 
barrasses the  beginner ;  nevertheless,  on  considera- 
tion, it  will  appear  to  be  very  similar  to  the  sense  in 
which  they  are  used  in  common  life.  We  do  not  form 
our  ideas  of  smallness  or  greatness  from  the  actual 
numbers  which  are  contained  in  a  collection,  but  from 
the  proportion  which  the  numbers  bear  to  those  which 


ROOTS  IN  GENERAL  AND  LOGARITHMS.      1 71 

are  usually  found  in  similar  collections.  Thus  of  1000 
men  we  should  say,  if  they  lived  in  one  village,  that 
it  was  extremely  large ;  if  they  formed  a  regiment, 
that  it  was  rather  large ;  if  an  army,  that  it  was  ut- 
terly insignificant  in  point  of  numbers.  Hence,  in 
such  an  expression  as  Ah  -f-  BW  -j-  Ch3,  we  may,  if  h  is 
very  small,  reject  BW  -f-  ChB,  as  being  very  small  com- 
pared with  Ah.  An  error  will  thus  be  committed,  but 
a  very  small  one  only,  and  which  becomes  smaller  as 
h  becomes  smaller. 

Let  us  take  any  algebraical  expression,  such  as 
x2  -{-  x,  and  suppose  that  x  is  increased  by  a  very  small 
quantity  h.  The  expression  then  becomes  (<#-f~^)2-|~ 
(x  +  h),  or  x2  +  x  +  (2x+l)A  +  tf.  But  it  was  x*  +  x; 
therefore,  in  consequence  of  x  receiving  the  increment 
h,  x2  -f-  x  has  received  the  increment  (2x-{-  \}h-\-h2t 
for  which  (2^-J-l)^  may  be  written,  since  h  is  very 
small.  This  is  proportional  to  h,  since,  if  h  were 
doubled,  (2#-|-l)^  would  be  doubled;  also,  if  the 
first  were  halved  the  second  would  be  halved,  etc.  In 
general,  if  y  is  a  quantity  which  contains  x,  and  if  x 
be  changed  into  x  -f-  h,  y  is  changed  into  a  quantity  of 
the  form  y  -f  Ah  +  Bh*  -f  Ch*  +  etc.;  that  is,  y  re- 
ceives an  increment  of  the  form  Ah  -j-  Bh^  -j-  Ch*  -j-  etc. 
If  h  be  very  small,  this  may,  without  sensible  error, 
be  reduced  to  its  first  term,  viz.,  Ah,  which  is  propor- 
tional to  h.  The  general  proof  of  this  proposition  be- 
longs to  a  higher  department  of  mathematics  ;  never- 
theless, the  student  may  observe  that  it  holds  good  in 


172  ON  THE  STUDY  OF  MATHEMATICS. 

all  the  instances  which  occur  in  elementary  treatises 
on  arithmetic  and  algebra. 
For  example  : 

=  xm  -f  m  xm~l  h  +  m        —  ^  xm~2  A2  +  etc. 


Here  A  =  m  xm-1,  B  =  m  —      -  xm~'2,  etc.;  and  if  h  be 

a 

very  small,  (x  -\-  hy  =  xm-\-  mxm~1A,  nearly. 

Again,  ^5«  1  +  *  +  *  +  '^  +  etc.     Therefore, 


e*  X  e*  or  £*+*  =  ex  +  e*h+        W  +  etc.     And  if  h  be 

a 

very  small,  c*+h  =  ex  -f  e*h,  nearly. 

Again,    log.  (1  +  »')  =  M(n'  —  1«'2  -f  J«'3—  etc.). 
To  each  side  add  log.*,  recollecting  that 
log.A:  +  log.  (1  +  »')  =  log.  *(1  +  «')  =  log.  (x  +  xtt), 

and  let 

,       x 

xn  =h  or  n  =  —. 
n 

Making  these  substitutions,  the  equation  becomes 

-fete. 


M 
If  h  is  very  small,  log.  (.#  -f-  /f)  =  log.  x-\-  "  -  h. 

3C 

We  can  now  apply  this  to  the  logarithmic  example 
with  which  we  commenced  this  subject.  It  appears 
that 

Log.  41713  =  4  .  6202714 

Log.  (41713  +  1)  =  4  .  6202714  +  .  0000104 
Log.  (41713  +  2)  =  4  .  6202714  +  .  0000104  X  2. 
From  which,   and  the  considerations  above-men- 
tioned, 


ROOTS  IN  GENERAL  AND  LOGARITHMS.       173 

Log.  (41713  -M)  =  log.  41713  -f  .  0000104  x  h, 
which  is  extremely  near  the  truth,  even  when  h  is  a 
much  larger  number,  as  the  tables  will  show.  Sup- 
pose, then,  that  the  logarithm  of  41713.27  is  required. 
Here  /$  =  .27.  It  therefore  only  remains  to  calculate 
.0000104  X- 27,  and  add  the  result,  or  as  much  of  it 
as  is  contained  in  the  first  seven  places  of  decimals, 
to  the  logarithm  of  41713.  This  trouble  is  saved  in 
the  tables  in  the  following  manner.  The  difference  of 
the  successive  logarithms  is  written  down,  with  the 
exception  of  the  cyphers  at  the  beginning,  in  the 
column  marked  D  or  Diff. ,  under  which  are  registered 
the  tenths  of  that  difference,  or  as  much  of  them  as  is 
contained  in  the  first  seven  decimal  places,  increasing 
the  seventh  figure  by  1  when  the  eighth  is  equal  to  or 
greater  than  5,  and  omitting  the  cyphers  to  save  room. 
From  this  table  of  tenths  the  table  of  hundredth  parts 
may  be  made  by  striking  off  the  last  figure,  making 
the  usual  change  in  the  last  but  one,  when  the  last  is 
equal  to  or  greater  than  5,  and  placing  an  additional 
cypher.  The  logarithm  of  41713.27  is,  therefore,  ob- 
tained in  the  following  manner  : 

Log.  41713  =4. 6202714 
. 0000104  X- 2  =  .0000021 
.  00001 04  x  •  07  =  .0000007 

Log.  41713.27  =4.6202742 

This,  when  the  useless  cyphers  and  parts  of  the  opera- 
tion are  omitted,  is  the  process  given  in  all  the  books 
of  logarithms.  If  the  logarithm  of  a  number  contain- 


174  ON  THE  STUDY  OF  MATHEMATICS. 

ing  more  than  seven  significant  figures  be  sought,  for 
example  219034.717,  recourse  must  be  had  to  a  table, 
in  which  the  logarithms  are  carried  to  more  than  seven 
places  of  decimals.  The  fact  is,  that  in  the  first  seven 
places  of  decimals  there  is  no  difference  between 
log.  219034. 7  and  log.  219034.717.  For  an  excellent 
treatise  on  the  practice  of  logarithms  the  reader  may 
consult  the  preface  to  Babbage's  Table  of  Logarithms  * 

*  Copies  of  Babbage's  Table  of  Logarithms  are  now  scarce,  and  the  reader 
may  accordingly  be  referred  to  the  prefaces  of  the  treatises  mentioned  no 
page  168.  The  article  on  "  Logarithms,  Use  of"  in  the  English  Cyclopedia, 
may  also  be  consulted  with  profit. — Ed. 


CHAPTER  XII. 

ON  THE  STUDY  OF  ALGEBRA. 

IN  this  chapter  we  shall  give  the  student  some  ad- 
vice as  to  the  manner  in  which  he  should  prose- 
cute his  studies  in  algebra.  The  remaining  parts  of 
this  subject  present  a  field  infinite  in  its  extent  and  in 
the  variety  of  the  applications  which  present  them- 
selves. By  whatever  name  the  remaining  parts  of 
the  subject  may  be  called,  even  though  the  ideas  on 
which  they  are  based  may  be  geometrical,  still  the 
mechanical  processes  are  algebraical,  and  present  con- 
tinual applications  of  the  preceding  rules  and  devel- 
opments of  the  subjects  already  treated.  This  is  the 
case  in  Trigonometry,  the  application  of  Algebra  to 
Geometry,  the  Differential  Calculus,  or  Fluxions,  etc. 
I.  The  first  thing  to  be  attended  to  in  reading  any 
algebraical  treatise,  is  the  gaining  a  perfect  under- 
standing of  the  different  processes  there  exhibited, 
and  of  their  connexion  with  one  another.  This  can- 
not be  attained  by  a  mere  reading  of  the  book,  how- 
ever great  the  attention  which  may  be  given.  It  is 


176  ON  THE  STUDY  OF  MATHEMATICS. 

impossible,  in  a  mathematical  work,  to  fill  up  every 
process  in  the  manner  in  which  it  must  be  filled  up  in 
the  mind  of  the  student  before  he  can  be  said  to  have 
completely  mastered  it.  Many  results  must  be  given, 
of  which  the  details  are  suppressed,  such  are  the  ad- 
ditions, multiplications,  extractions  of  the  square  root, 
etc.,  with  which  the  investigations  abound.  These 
must  not  be  taken  on  trust  by  the  student,  but  must 
be  worked  by  his  own  pen,  which  must  never  be  out 
of  his  hand  while  engaged  in  any  algebraical  process. 
The  method  which  we  recommend  is,  to  write  the 
whole  of  the  symbolical  part  of  each  investigation, 
filling  up  the  parts  to  which  we  have  alluded,  adding 
only  so  much  verbal  elucidation  as  is  absolutely  neces- 
sary to  explain  the  connexion  of  the  different  steps, 
which  will  generally  be  much  less  than  what  is  given 
in  the  book.  This  may  appear  an  alarming  labor  to 
one  who  has  not  tried  it,  nevertheless  we  are  con- 
vinced that  it  is  by  far  the  shortest  method  of  pro- 
ceeding, since  the  deliberate  consideration  which  the 
act  of  writing  forces  us  to  give,  will  prevent  the  con- 
fusion and  difficulties  which  cannot  fail  to  embarrass 
the  beginner  if  he  attempt,  by  mere  perusal  only,  to 
understand  new  reasoning  expressed  in  new  language. 
If,  while  proceeding  in  this  manner,  any  difficulty 
should  occur,  it  should  be  written  at  full  length,  and 
it  will  often  happen  that  the  misconception  which  oc- 
casioned the  embarrassment  will  not  stand  the  trial  to 
which  it  is  thus  brought.  Should  there  be  still  any 


THE  STUDY  OP  ALGEBRA.  177 

matter  of  doubt  which  is  not  removed  by  attentive  re- 
consideration, the  student  should  proceed,  first  mak- 
ing a  note  of  the  point  which  he  is  unable  to  perceive. 
To  this  he  should  recur  in  his  subsequent  progress, 
whenever  he  arrives  at  anything  which  appears  to 
have  any  affinity,  however  remote,  to  the  difficulty 
which  stopped  him,  and  thus  he  will  frequently  find 
himself  in  a  condition  to  decypher  what  formerly 
appeared  incomprehensible.  In  reasoning  purely  geo- 
metrical, there  is  less  necessity  for  committing  to  writ- 
ing the  whole  detail  of  the  arguments,  since  the  sym- 
bolical language  is  more  quickly  understood,  and  the 
subject  is  in  a  great  measure  independent  of  the  mech- 
anism of  operations  ;  but,  in  the  processes  of  algebra, 
there  is  no  point  on  which  so  much  depends,  or  on 
which  it  becomes  an  instructor  more  strongly  to  in- 
sist. 

II.  On  arriving  at  any  new  rule  or  process,  the 
student  should  work  a  number  of  examples  sufficient 
to  prove  to  himself  that  he  understands  and  can  apply 
the  rule  or  process  in  question.  Here  a  difficulty  will 
occur,  since  there  are  many  of  these  in  the  books,  to 
which  no  examples  are  formally  given.  Nevertheless, 
he  may  choose  an  example  for  himself,  and  his  pre- 
vious knowledge  will  suggest  some  method  of  proving 
whether  his  result  is  true  or  not.  For  example,  the 
development  of  (a-\-x)"$  will  exercise  him  in  the  use 
of  the  binomial  theorem  ;  when  he  has  obtained  the 

7 

series  which  is  equivalent  to  (#-|-#)3,  let  him,  in  the 


178  ON  THE  STUDY  OF  MATHEMATICS. 

same  way,  develop  (a-\-x^'}  the  product  of  these, 
since  |-|-|=r3,  ought  to  be  the  same  as  the  develop- 
ment of  (a  +  x)*,  or  as  a9  -f  3a*  x  -\-3ax*  +  x*.  He 
may  also  try  whether  the  development  of  (a-\-x^  by 
the  binomial  theorem,  gives  the  same  result  as  is  ob- 
tained by  the  extraction  of  the  square  root  of  a  -|-  A\ 
Again,  when  any  development  is  obtained,  it  should 
be  seen  whether  the  development  *p°ssesses  aU  the 
properties  of  the  expression  from  which  it  has  been 
derived.  For  example,  -  —  —  is  proved  to  be  equiv- 

-1    •      OC 

alent  to  the  series 

1  -j-  x  -\-  x2  -\-  xs  +,  etc.  ,  ad  infinitum. 

This,  when  multiplied  by  1  —  x,  should  give  1  ;  when 
multiplied  by  1  —  x2,  should  give  \-\-x,  because 

-J—X(1  —*)=!,   r-  —  X(l  —  x^  =  l  +  x,  etc. 

J-  -       OC  -L  -  OC 

Again, 


3        +...admf. 


etc. 


Now,  since  a*  X  a-y  =  a*fj',  the  product  of  the  two 
first  series  should  give  the  third.  Many  other  in- 
stances of  the  same  sort  will  suggest  themselves,  and 
a  careful  attention  to  them  will  confirm  the  demon- 
stration of  the  several  theorems,  which,  to  a  beginner, 


THE  STUDY  OF  ALGEBRA.  179 

is  often  doubtful,  on  account  of  the  generality  of  the 
reasoning. 

III.  Whenever  a  demonstration  appears  perplexed, 
on  account  of  the  number  and  generality  of  the  sym- 
bols, let  some  particular  case  be  chosen,  and  let  the 
same  demonstration  be  applied.  For  example,  if  the 
binomial  theorem  should  not  appear  sufficiently  plain, 
the  same  reasoning  may  be  applied  to  the  expansion 
of  (1  -f-^c)3,  or  any  other  case,  which  is  there  applied 
to  (1  -|-  #)*".  Again,  the  general  form  of  the  product 
(x  +  a),  (#-(-£),  (x-\-c),  etc.,  .  .  .  containing  n  factors, 
will  be  made  apparent  by  taking  first  two,  then  three, 
and  four  factors,  before  attempting  to  apply  the  rea- 
soning which  establishes  the  form  of  the  general  pro- 
duct. The  same  applies  particularly  to  the  theory  of 
permutations  and  combinations,  and  to  the  doctrine 
of  probabilities,  which  is  so  materially  connected  with 
it.  In  the  theory  of  equations  it  will  be  advisable  at 
first,  instead  of  taking  the  general  equation  of  the 
form 

+  M=  0, 


to  choose  that  of  the  third,  or  at  most  of  the  fourth 
degree,  or  both,  on  which  to  demonstrate  all  the 
properties  of  expressions  of  this  description.  But  in 
all  these  cases,  when  the  particular  instances  have 
been  treated,  the  general  case  should  not  be  neglected, 
since  the  power  of  reasoning  upon  expressions  such 
as  the  one  just  given,  in  which  all  the  terms  cannot 


l8o  ON  THE  STUDY  OF  MATHEMATICS. 

be  written  down,  on  account  of  their  indeterminate 
number,  must  be  exercised,  before  the  student  can 
proceed  with  any  prospect  of  success  to  the  higher 
branches  of  mathematics. 

IV.  When  any  previous  theorem  is  referred  to,  the 
reference   should   be  made,    and  the  student  should 
satisfy  himself  that  he  has  not  forgotten  its  demon- 
stration.    If  he  finds  that  he  has  done  so,  he  should 
not  grudge  the  time  necessary  for  its  recovery.     By 
so  doing,  he  will  avoid  the  necessity  of  reading  over 
the  subject  again,  and  will  obtain  the  additional  ad- 
vantage of  being  able  to  give  to  each  part  of  the  sub- 
ject a  time   nearly  proportional   to   its   importance, 
whereas,  by  reading  a  book  over  and  over  again  until 
he  is  a  master  of  it,  he  will  not  collect  the  more  prom- 
inent parts,  and  will  waste  time  upon  unimportant 
details,  from  which  even  the  best  books  are  not  free. 
The  necessity  for  this  continual  reference  is  particu- 
larly felt  in  the  Elements  of  Geometry,  where  allusion 
is   constantly  made   to  preceding   propositions,    and 
where  many  theorems  are  of  no  importance,  consid- 
ered as  results,  and  are  merely  established  in  order  to 
serve  as  the  basis  of  future  propositions. 

V.  The  student  should  not  lose  any  opportunity 
of   exercising  himself  in  numerical  calculation,  and 
particularly  in  the  use  of  the  logarithmic  tables.     His 
power  of  applying  mathematics  to  questions  of  prac- 
tical utility  is  in  direct  proportion  to  the  facility  which 
he  possesses  in  computation.     Though  it  is  in  plane 


THE  STUDY  OF  ALGEBRA.  l8l 

and  spherical  trigonometry  that  the  most  direct  nu- 
merical applications  present  themselves,  nevertheless 
the  elementary  parts  of  algebra  abound  with  useful 
practical  questions.  Such  will  be  found  resulting  from 
the  binomial  theorem,  the  theory  of  logarithms,  and 
that  of  continued  fractions.  The  first  requisite  in  this 
branch  of  the  subject,  is  a  perfect  acquaintance  with 
the  arithmetic  of  decimal  fractions  ;  such  a  degree  of 
acquaintance  as  can  only  be  gained  by  a  knowledge 
of  the  principles  as  well  as  of  the  rules  which  are  de- 
duced from  them.  From  the  imperfect  manner  in 
which  arithmetic  is  usually  taught,  the  student  ought 
in  most  cases  to  recommence  this  study  before  pro- 
ceeding to  the  practice  of  logarithms. 

VI.  The  greatest  difficulty,  in  fact  almost  the  only 
one  of  any  importance  which  algebra  offers  to  the  rea- 
son, is  the  use  of  the  isolated  negative  sign  in  such 
expressions  as  —  a,  a~x,  and  the  symbols  which  we 
have  called  imaginary.  It  is  a  remarkable  fact,  that 
the  first  elements  of  the  mathematics,  sciences  which 
demonstrate  their  results  with  more  certainty  than  any 
others,  contain  difficulties  which  have  been  the  sub- 
jects of  discussion  for  centuries.  In  geometry,  for 
example,  the  theory  of  parallel  lines  has  never  yet 
been  freed  from  the  difficulty  which  presented  itself  to 
Euclid,  and  obliged  him  to  assume,  instead  of  proving, 
the  12th  axiom  of  his  first  book.  Innumerable  as  have 
been  the  attempts  to  elude  or  surmount  this  obstacle, 
no  one  has  been  more  successful  than  another.  The 


I  82  ON  THE  STUDY  OF  MATHEMATICS. 

elements  of  fluxions  or  the  differential  calculus,  of 
mechanics,  of  optics,  and  of  all  the  other  sciences,  in 
the  same  manner  contain  difficulties  peculiar  to  them- 
selves. These  are  not  such  as  would  suggest  them- 
selves to  the  beginner,  who  is  usually  embarrassed  by 
the  actual  performance  of  the  operations,  and  no  ways 
perplexed  by  any  doubts  as  to  the  foundations  of  the 
rules  by  which  he  is  to  work.  It  is  the  characteristic 
of  a  young  student  in  the  mathematical  sciences,  that 
he  sees,  or  fancies  that  he  sees,  the  truth  of  every  re- 
sult which  can  be  stated  in  a  few  words,  or  arrived  at 
by  few  and  simple  operations,  while  that  which  is  long 
is  always  considered  by  him  as  abstruse.  Thus  while 
he  feels  no  embarrassment  as  to  the  meaning  of  the 
equation  -f-  a  X  — 0  =  —  #2>  he  considers  the  multipli- 
cation of  am  -\-  an  by  bm  -f  bn  as  one  of  the  difficulties 
of  algebra.  This  arises,  in  our  opinion,  from  the  man- 
ner in  which  his  previous  studies  are  usually  con- 
ducted. From  his  earliest  infancy,  he  learns  no  fact 
from  his  own  observation,  he  deduces  no  truth  by  the 
exercise  of  his  own  reason.  Even  the  tables  of  arith- 
metic, which,  with  a  little  thought  and  calculation,  he 
might  construct  for  himself,  are  presented  to  him 
ready  made,  and  it  is  considered  sufficient  to  commit 
them  to  memory.  Thus  a  habit  of  examination  is  not 
formed,  and  the  student  comes  to  the  science  of  alge- 
bra fully  prepared  to  believe  in  the  truth  of  any  rule 
which  is  set  before  him,  without  other  authority  than 
the  fact  of  finding  it  in  the  book  to  which  he  is  recom- 


THE  STUDY  OF  ALGEBRA.  183 

mended.  It  is  no  wonder,  then,  that  he  considers  the 
difficulty  of  a  process  as  proportional  to  that  of  re- 
membering and  applying  the  rule  which  is  given, 
without  taking  into  consideration  the  nature  of  the 
reasoning  on  which  the  rule  was  founded.  We  are 
not  advocates  for  stopping  the  progress  of  the  student 
by  entering  fully  into  all  the  arguments  for  and  against 
such  questions,  as  the  use  of  negative  quantities,  etc., 
which  he  could  not  understand,  and  which  are  incon- 
clusive on  both  sides ;  but  he  might  be  made  aware 
that  a  difficulty  does  exist,  the  nature  of  which  might 
be  pointed  out  to  him,  and  he  might  then,  by  the  con- 
sideration of  a  sufficient  number  of  examples,  treated 
separately,  acquire  confidence  in  the  results  to  which 
the  rules  lead.  Whatever  may  be  thought  of  this 
method,  -it  must  be  better  than  an  unsupported  rule, 
such  as  is  given  in  many  works  on  algebra. 

It  may  perhaps  be  objected  that  this  is  induction, 
a  species  of  reasoning  which  is  foreign  to  the  usually 
received  notions  of  mathematics.  To  this  it  may  be 
answered,  that  inductive  reasoning  is  of  as  frequent 
occurrence  in  the  sciences  as  any  other.  It  is  certain 
that  most  great  discoveries  have  been  made  by  means 
of  it ;  and  the  mathematician  knows  that  one  of  his 
most  powerful  engines  of  demonstration  is  that  pecu- 
liar species  of  induction  which  proves  many  general 
truths  by  demonstrating  that,  if  the  theorem  be  true 
in  one  case,  it  is  true  for  the  succeeding  one.  But  the 
beginner  is  obliged  to  content  himself  with  a  less  rig- 


184  ON  THE  STUDY  OF  MATHEMATICS. 

orous  species  of  proof,  though  equally  conclusive,  as 
far  as  moral  certainty  is  concerned.  Unable  to  grasp 
the  generalisations  with  which  the  more  advanced 
student  is  familiar,  he  must  satisfy  himself  of  the 
truth  of  general  theorems  by  observing  a  number  of 
particular  simple  instances  which  he  is  able  to  com- 
prehend. For  example,  we  would  ask  any  one  who 
has  gone  over  this  ground,  whether  he  derived  more 
certainty  as  to  the  truth  of  the  binomial  theorem  from 
the  general  demonstration  (if  indeed  he  was  suffered 
to  see  it  so  early  in  his  career) ,  or  from  observation 
of  its  truth  in  the  particular  cases  of  the  development 
of  (#-|-^)2>  (#-M)3>  etc-»  substantiated  by  ordinary 
multiplication.  We  believe  firmly,  that  to  the  mass 
of  young  students,  general  demonstrations  afford  no 
conviction  whatever;  and  that  the  same  may  be  said 
of  almost  every  species  of  mathematical  reasoning, 
when  it  is  entirely  new.  We  have  before  observed, 
that  it  is  necessary  to  learn  to  reason ;  and  in  no  case 
is  the  assertion  more  completely  verified  than  in  the 
study  of  algebra.  It  was  probably  the  experience  of 
the  inutility  of  general  demonstrations  to  the  very 
young  student  that  caused  the  abandonment  of  rea- 
soning which  prevailed  so  much  in  English  works  on 
elementary  mathematics.  Rules  which  the  student 
could  follow  in  practice  supplied  the  place  of  argu- 
ments which  he  could  not,  and  no  pains  appear  to 
have  been  taken  to  adopt  a  middle  course,  by  suiting 
the  nature  of  the  proof  to  the  student's  capacity.  The 


THE  STUDY  OF  ALGEBRA.  185 

objection  to  this  appears  to  have  been  the  necessity 
which  arose  for  departing  from  the  appearance  of  rig- 
orous demonstration.  This  was  the  cry  of  those  who, 
not  having  seized  the  spirit  of  the  processes  which 
they  followed,  placed  the  force  of  the  reasoning  in  the 
forms.  To  such  the  authority  of  great  names  is  a 
strong  argument;  we  will  therefore  cite  the  words  of 
Laplace  on  this  subject. 

"Newton  extended  to  fractional  and  negative 
powers  the  analytical  expression  which  he  had  found 
for  whole  and  positive  ones.  You  see  in  this  exten- 
sion one  of  the  great  advantages  of  algebraic  language 
which  expresses  truths  much  more  general  than  those 
which  were  at  first  contemplated,  so  that  by  making 
the  extension  of  which  it  admits,  there  arises  a  multi- 
tude of  new  truths  out  of  formulae  which  were  founded 
upon  very  limited  suppositions.  At  first,  people  were 
afraid  to  admit  the  general  consequences  with  which 
analytical  formulae  furnished  them  ;  but  a  great  number 
of  examples  having  verified  them,  we  now,  without  fear, 
yield  ourselves  to  the  guidance  of  analysis  through  all 
the  consequences  to  which  it  leads  us,  and  the  most 
happy  discoveries  have  sprung  from  the  boldness. 
We  must  observe,  however,  that  precautions  should 
be  taken  to  avoid  giving  to  formulae  a  greater  exten- 
sion than  they  really  admit,  and  that  it  is  always  well 
to  demonstrate  rigorously  the  results  which  are  ob- 
tained." 

have  observed  that  beginners  are  not  disposed 


1 86  ON  THE  STUDY  OF  MATHEMATICS. 

to  quarrel  with  a  rule  which  is  easy  in  practice,  and 
verified  by  examples,  on  account  of  difficulties  which 
occur  in  its  establishment.  The  early  history  of  the 
sciences  presents  occasion  for  the  same  remark.  In 
the  work  of  Diophantus,  the  first  Greek  writer  on  al- 
gebra, we  find  a  principle  equivalent  to  the  equations 
+  #X — b  = — ab,  and  — «X — b  =  -{-ab,  admitted 
as  an  axiom,  without  proof  or  difficulty.  In  the  Hin- 
doo works  on  algebra,  and  the  Persian  commentators 
upon  them,  the  same  thing  takes  place.  It  appears, 
that  struck  with  the  practical  utility  of  the  rule,  and 
certain  by  induction  of  its  truth,  they  did  not  scruple 
to  avail  themselves  of  it.  A  more  cultivated  age,  pos- 
sessed of  many  formulae  whose  developments  pre- 
sented striking  examples  of  an  universality  in  alge- 
braic language  not  contemplated  by  its  framers,  set 
itself  to  inquire  more  closely  into  the  first  principles 
of  the  science.  Long  and  still  unfinished  discussions 
have  been  the  result,  but  the  progress  of  nations  has 
exhibited  throughout  a  strong  resemblance  to  that  of 
individuals. 

VII.  The  student  should  make  for  himself  a  sylla- 
bus of  results  only,  unaccompanied  by  any  demonstra- 
tion. It  is  essential  to  acquire  a  correct  memory  for 
algebraical  formulae,  which  will  save  much  time  and 
labor  in  the  higher  departments  of  the  science.  Such 
a  syllabus  will  be  a  great  assistance  in  this  respect, 
and  care  should  be  taken  that  it  contain  only  the  most 
useful  and  most  prominent  formulae.  Whenever  that 


THE  STUDY  OF  ALGEBRA.  187 

can  be  done,  the  student  should  have  recourse  to  the 
system  of  tabulation,  of  which  he  will  have  seen  sev- 
eral examples  in  this  treatise.  In  this  way  he  should 
write  the  various  forms  which  the  roots  of  the  equa- 
tion ax2-\-  bx  +  £  =  0  assume,  according  to  the  signs 
of  a,  b,  and  <:,  etc.  Both  the  preceptor  and  the  pupil, 
but  especially  the  former,  will  derive  great  advantage 
from  the  perusal  of  Lacroix,  Essais  sur  PEnseignement 
'en  general  et  sur  celui  des  Mathe'matiques  en  particulier* 
Condillac,  La  Langue  des  Calculs,  and  the  various  ar- 
ticles on  the  elements  of  algebra  in  the  French  En- 
cyclopedia, which  are  for  the  most  part  written  by 
D'Alembert.  The  reader  will  here  find  the  first  prin- 

*The  books  mentioned  in  the  present  passage,  while  still  very  valuable, 
are  now  not  easily  procurable  and,  besides,  do  not  give  a  complete  idea  of 
the  subject  in  its  modern  extent.  A  recent  work  on  the  Philosophy  and  Teach- 
ing of  Mathematics  is  that  of  C.  A.  Laisant  (La  Mathtmatique.  Philosophie- 
Enseignement.  Paris,  1898,  Georges  Carre  et  C.  Naud,  publishers.)  Perhaps 
the  most  accessible  and  useful  work  in  English  for  the  elements  is  David 
Eugene  Smith's  new  book  The  Teaching  of  Elementary  Mathematics.  (New 
York  :  The  Macmillan  Company,  1900).  Mention  might  be  made  also  of  W.  M. 
Gillespie's  translation  from  Comte's  Cours  de  Philosophie  Positive,  under  the 
title  of  The  Philosophy  of  Mathematics  (New  York  :  Harpers,  1851),  and  of  the 
Cours  de  Methodologie  MathSmatique  of  Felix  Dauge  (Deuxieme  edition,  revue 
et  augmented.  Gand,  Ad.  Hoste  ;  Paris,  Gauthier-Villars,  1896).  The  recent 
work  of  Freycinet  on  the  Philosophy  of  the  Sciences  (Paris,  1896,  Gauthier-Vil- 
lars) will  be  found  valuable.  One  of  the  best  and  most  comprehensive  of  the 
modern  works  is  that  of  Duhamel,  Des  M&thodes  dans  les  Sciences  de  Raisonne- 
ment,  (5  parts,  Paris,  Gauthier-Villars),  a  work  giving  a  comprehensive  expo- 
sition of  the  foundations  of  all  the  mathematical  sciences.  The  chapters  in 
Diihring's  Kritische  Geschichte  der  Prinzipien  der  Mechanik  and  his  Neue 
Grundmittel  on  the  study  of  mathematics  and  mechanics  is  replete  with  orig- 
inal, but  hazardous,  advice,  and  may  be  consulted  as  a  counter-irritant  to 
the  traditional  professional  views  of  the  subject.  The  articles  in  the  English 
Cyclopcedia,  by  DeMorgan  himself,  contain  refreshing  hints  on  this  subject. 
But  the  greatest  inspiration  is  to  be  drawn  from  the  works  of  the  masters 
themselves;  for  example,  from  such  works  as  Laplace's  Introduction  to  the 
Calculus  of  Probabilities,  or  from  the  historical  and  philosophical  reflexions 
that  uniformly  accompany  the  later  works  of  Lagrange.  The  same  remark 
applies  to  the  later  mathematicians  of  note. — Ed. 


1 88  ON  THE  STUDY  OF  MATHEMATICS. 

ciples  of  algebra,  developed  and  elucidated  in  a  mas- 
terly manner.  A  great  collection  of  examples  will  be 
found  in  most  elementary  works,  but  particularly  in 
Hirsch,  Sammlung  von  Beispielen,  etc.,  translated  into 
English  under  the  title  of  Self -Examinations  in  Algebra, 
etc.,  London:  Black,  Young  and  Young,  1825.*  The 
student  who  desires  to  carry  his  algebraical  studies 
farther  than  usual,  and  to  make  them  the  stepping- 
stone  to  a  knowledge  of  the  higher  mathematics, 
should  be  acquainted  with  the  French  language,  f  A 
knowledge  of  this,  sufficient  to  enable  him  to  read  the 
simple  and  easy  style  in  which  the  writers  of  that  na- 
tion treat  the  first  principles  of  every  subject,  may  be 
acquired  in  a  short  time.  When  that  is  done,  we  re- 
commend to  the  student  the  algebra  of  M.  Bourdon,  J 

*Hirsch's  Collection,  enlarged  and  modernised,  can  be  obtained  in  vari- 
ous recent  German  editions.  The  old  English  translations  of  the  original 
are  not  easily  procured. — Ed. 

t  German  is  now  of  as  much  importance  as  French.  But  the  French  text- 
books still  retain  their  high  standard. — Ed. 

\  Bourdon's  Elements  of  Algebra  is  still  used  in  France,  having  appeared 
in  1895  in  its  eighteenth  edition,  with  notes  by  M.  Prouhet  (Gauthier-Villars, 
Paris.)  A  more  elementary  French  work  of  a  modern  character  is  that  of 
J.  Collin  (Second  edition,  1888,  Paris,  Gauthier-Villars).  A  larger  and  more' 
complete  treatise  which  begins  with  the  elements  and  extends  to  the  higher 
branches  of  the  subject  is  the  Traitf  d'Algtbre  of  H.  Laurent,  in  four  small 
volumes  (Gauthier-Villars,  Paris).  This  work  contains  a  large  collection  of 
examples.  Another  elementary  work  is  that  of  C.  Bourlet,  Lecons  d' Algtbre 
Elementaire,  Paris,  Colin,  1896.  A  standard  and  exhaustive  work  on  higher 
algebra  is  the  Cours  d'Algebre  Suptrieure,  of  J.  A.  Serret,  two  large  volumes 
(Fifth  edition,  1885,  Paris,  Gauthier-Villars). 

The  number  of  American  and  English  text-books  of  the  intermediate  and 
higher  type  is  very  large.  Todhunter's  Afgebra  and  Theory  of  Equations 
(London:  Macmillan  &  Co.)  were  for  a  long  time  the  standards  in  England 
and  this  country,  but  have  now  (especially  the  first-mentioned)  been  virtually 
superseded.  An  excellent  recent  text-book  for  beginners,  and  one  that  skil- 
fully introduces  modern  notions,  is  the  Elements  of  Algebra  of  W.  W.  Beman 
and  P.  E.  Smith  (Boston,  1900).  Fisher  and  Schwatt's  elementary  text-books 


THE  STUDY  OF  ALGEBRA.  1 89 

a  work  of  eminent  merit,  though  of  some  difficulty  to 
the  English  student,  and  requiring  some  previous 
habits  of  algebraical  reasoning. 

VIII.  The  height  to  which  algebraical  studies 
should  be  carried,  must  depend  upon  the  purpose  to 
which  they  are  to  be  applied.  For  the  ordinary  pur- 
poses of  practical  mathematics,  algebra  is  principally 
useful  as  the  guide  to  trigonometry,  logarithms,  and 
the  solution  of  equations.  Much  and  profound  study 


of  algebra  are  also  recommendable  from  both  a  practical  and  theoretical 
point  of  view.  Valuable  are  C.  Smith's  Treatise  on  Algeb ra  (London  :  Mac- 
millan),  and  Oliver,  Wait,  and  Jones's  Treatise  on  Algebra  (Ithaca,  N.  Y., 
1887),  also  Fine's  Number  System  of  Algebra  (Boston  :  Leach).  The  best  Eng- 
lish work  on  the  theory  of  equations  is  Burnside  and  Panton's  (Longmans). 

A  very  exhaustive  presentation  of  the  subject  from  the  modern  point 
of  view  is  the  Algebra  of  Professor  George  Chrystal  (Edinburgh:  Adam  and 
Charles  Black,  publishers),  in  two  large  volumes  of  nearly  six  hundred  pages 
each.  Recently  Professor  Chrystal  has  published  a  more  elementary  work 
entitled  Introduction  to  Algebra  (same  publishers). 

A  few  German  works  may  also  be  mentioned  in  this  connexion,  for  the 
benefit  of  readers  acquainted  with  that  language.  Professor  Hermann  Schu- 
bert has,  in  various  forms,  given  systematic  expositions  of  the  elementary 
principles  of  arithmetic,  (e.  g.,  see  his  Arithmetik  und  Algebra,  Sammlung 
Goschen,  Leipsic,— an  extremely  cheap  series  containing  several  other  ele- 
mentary mathematical  works  of  high  standard ;  also,  for  a  statement  of 
Schubert's  views  in  English  consult  his  Mathematical  Recreations,  Chicago, 
1898).  Professor  Schubert  has  recently  begun  the  editing  of  a  new  and  larger 
series  of  mathematical  text-books  called  the  Sammlung  Schubert  (Leipsic : 
Goschen),  which  contains  three  works  treating  of  algebra.  In  this  connexion 
maybe  mentioned  also  Matthiessen's  admirable  Grundziige  der  antiken  und 
modernen  Algebra  (Leipsic  :  Teubner)  for  literal  equations.  The  following 
are  all  excellent:  (i)  Otto  Biermann's  Elemente  der  hb'heren  Mathematik 
(Leipsic,  1895);  (2)  Petersen's  Theorie  der  algebraischen  Gleichungen  (Copen- 
hagen: Host;  also  in  French,  Paris  :  Gauthier-Villars);  (3)  Richard  Baltzer's 
Elemente  der  Mathematik  (2  vols.,  Leipsic:  Hirzel);  (4)  Gustav  Holzmiiller's 
Methodisches Lehrbuch  der  Elementarmathematik  (3  parts,  Leipsic:  Teubner); 

(5)  Werner  Jos.  Schiiller's  Arithmetik  und  Algebra  fur  hb'here  Schulen  und 
Lehrerseminare,  besonders  zum  Selbstunterricht ,  etc.  (Leipsic,  1891,  Teubner); 

(6)  Oskar  Schlomilch's  Handbuch  der  algebraischen  Analysis  (Frommann, 
Stuttgart);  (7)  Eugen  Netto's  Vorlesungen  uber  Algebra  (Leipsic :  Teubner,  2 
vols.);  (8)  Heinrich  Weber's  Lehrbuch  der  Algebra  (Braunschweig:  Vieweg,  2 
vols  ).    This  last  work  is  the  most  advanced  treatise  that  has  yet  appeared. 
A  French  translation  has  been  announced. — Ed. — April,  1902. 


I QO  ON  THE  STUDY  OF  MATHEMATICS. 

is  not  therefore  requisite ;  the  student  should  pay 
great  attention  to  all  numerical  processes  and  particu- 
larly to  the  methods  of  approximation  which  he  will 
find  in  all  the  books.  His  principal  instrument  is  the 
table  of  logarithms  of  which  he  should  secure  a  knowl- 
edge both  theoretical  and  practical.  The  course  which 
should  be  adopted  preparatory  to  proceeding  to  the 
higher  branches  of  mathematics  is  different.  It  is  still 
of  great  importance  that  the  student  should  be  well 
acquainted  with  numerical  applications  ;  nevertheless, 
he  may  omit  with  advantage  many  details  relative  to 
the  obtaining  of  approximative  numerical  results,  par- 
ticularly in  the  theory  of  equations  of  higher  degrees 
than  the  second.  Instead  of  occupying  himself  upon 
these,  he  should  proceed  to  the  application  of  algebra 
to  geometry,  and  afterwards  to  the  differential  cal- 
culus. When  a  competent  knowledge  of  these  has 
been  obtained,  he  may  then  revert  to  the  subjects 
which  he  has  neglected,  giving  them  more  or  less  at- 
tention according  to  his  own  opinion  of  the  use  which 
he  is  likely  to  have  for  them.  This  applies  particu- 
larly to  the  theory  of  equations,  which  abounds  with 
processes  of  which  very  few  students  will  afterwards 
find  the  necessity. 

We  shall  proceed  in  the  next  number  to  the  diffi- 
culties which  arise  in  the  study  of  Geometry  and  Tri- 
gonometry. 


CHAPTER  XIII. 

ON  THE  DEFINITIONS  OF  GEOMETRY. 

IN  this  treatise  on  the  difficulties  of  Geometry  and 
Trigonometry,  we  propose,  as  in  the  former  part 
of  the  work,  to  touch  on  those  points  only  which,  from 
novelty  in  their  principle,  are  found  to  present  diffi- 
culties to  the  student,  and  which  are  frequently  not 
sufficiently  dwelt  upon  in  elementary  works.  Perhaps 
it  may  be  asserted,  that  there  are  no  difficulties  in 
geometry  which  are  likely  to  place  a  serious  obstacle 
in  the  way  of  an  intelligent  beginner,  except  the  tem- 
porary embarrassment  which  always  attends  the  com- 
mencement of  a  new  study ;  that,  for  example,  there 
is  nothing  in  the  elements  of  pure  geometry  compar- 
able, in  point  of  complexity,  to  the  theory  of  the  nega- 
tive sign,  of  fractional  indices,  or  of  the  decomposi- 
tion of  an  expression  of  the  second  degree  into  factors. 
This  may  be  true  ;  and  were  it  only  necessary  to  study 
the  elements  of  this  science  for  themselves,  without 
reference  to  their  application,  by  means  of  algebra,  to 
higher  branches  of  knowledge,  we  should  not  have 


IQ2  ON  THE  STUDY  OF  MATHEMATICS. 

thought  it  necessary  to  call  the  attention  of  our  read- 
ers to  the  points  which  we  shall  proceed  to  place  be- 
fore them.  But  while  there  is  a  higher  study  in  which 
elementary  ideas,  simple  enough  in  their  first  form, 
are  so  generalised  as  to  become  difficult,  it  will  be  an 
assistance  to  the  beginner  who  intends  to  proceed 
through  a  wider  course  of  pure  mathematics  than 
forms  part  of  common  education,  if  his  attention  is 
early  directed,  in  a  manner  which  he  can  compre- 
hend, to  future  extensions  of  what  is  before  him. 

The  reason  why  geometry  is  not  so  difficult  as  al- 
gebra, is  to  be  found  in  the  less  general  nature  of  the 
symbols  employed.  In  algebra  a  general  proposition 
respecting  numbers  is  to  be  proved.  Letters  are  taken 
which  may  represent  any  of  the  numbers  in  question, 
and  the  course  of  the  demonstration,  far  from  making 
any  use  of  a  particular  case,  does  not  even  allow  that 
any  reasoning,  however  general  in  its  nature,  is  con- 
clusive, unless  the  symbols  are  as  general  as  the  argu- 
ments. We  do  not  say  that  it  would  be  contrary  to 
good  logic  to  form  general  conclusions  from  reasoning 
on  one  particular  case,  when  it  is  evident  that  the 
same  considerations  might  be  applied  to  any  other, 
but  only  that  very  great  caution,  more  than  a  beginner 
can  see  the  value  of,  would  be  requisite  in  deducing 
the  conclusion.  There  occurs  also  a  mixture  of  gen- 
eral and  particular  propositions,  and  the  latter  are 
liable  to  be  mistaken  for  the  former.  In  geometry  on 
the  contrary,  at  least  in  the  elementary  parts,  any 


THE  DEFINITIONS  OF  GEOMETRY.  IQ3 

proposition  may  be  safely  demonstrated  by  reasonings 
on  any  one  particular  example.  For  though  in  prov- 
ing a  property  of  a  triangle  many  truths  regarding 
that  triangle  may  be  asserted  as  having  been  proved 
before,  none  are  brought  forward  which  are  not  gen- 
eral, that  is,  true  for  all  instances  of  the  same  kind. 
It  also  affords  some  facility  that  the  results  of  elemen- 
tary geometry  are  in  many  cases  sufficiently  evident 
of  themselves  to  the  eye  ;  for  instance,  that  two  sides 
of  a  triangle  are  greater  than  the  third,  whereas  in 
algebra  many  rudimentary  propositions  derive  no  evi- 
dence from  the  senses;  for  example,  that  a3  —  b*  is 
always  divisible  without  remainder  by  a  —  b. 

The  definitions  of  the  simple  terms  point,  line,  and 
surface  have  given  rise  to  much  discussion.  But  the 
difficulties  which  attend  them  are  not  of  a  nature  to 
embarrass  the  beginner,  provided  he  will  rest  content 
with  the  notions  which  he  has  already  derived  from 
observation.  No  explanation  can  make  these  terms 
more  intelligible.  To  them  may  be  added  the  words 
straight  line,  which  cannot  be  mistaken  for  one  mo- 
ment, unless  it  be  by  means  of  the  attempt  to  explain 
them  by  saying  that  a  straight  line  is  "  that  which  lies 
evenly  between  its  extreme  points." 

The  line  and  surface  are  distinct  species  of  magni- 
tude, as  much  so  as  the  yard  and  the  acre.  The  first 
is  no  part  of  the  second,  that  is,  no  number  of  lines 
can  make  a  surface.  When  therefore  a  surface  is  di- 
vided into  two  parts  by  a  line,  the  dividing  line  is  not 


1Q4  ON  THE  STUDY  OF  MATHEMATICS. 

to  be  considered  as  forming  a  part  of  either.  That 
the  idea  of  the  line  or  boundary  necessarily  enters 
into  the  notion  of  the  division  is  very  true  ;  but  if  we 
conceive  the  line  abstracted,  and  thus  get  rid  of  the 
idea  of  division,  neither  surface  is  increased  or  dimin- 
ished, which  is  what  we  'mean  when  we  say  that  the 
line  is  not  a  part  of  the  surface.  The  same  considera- 
tions apply  to  a  point,  considered  as  the  boundary  of 
the  divisions  of  a  line. 

The  beginner  may  perhaps  imagine  that  a  line  is 
made  up  of  points,  that  is,  that  every  line  is  the  sum 
of  a  number  of  points,  a  surface  the  sum  of  a  number 
of  lines,  and  so  on.  This  arises  from  the  fact,  that 
the  things  which  we  draw  on  paper  as  the  representa- 
tives of  lines  and  points,  have  in  reality  three  dimen- 
sions, two  of  which,  length  and  breadth,  are  perfectly 
visible.  Thus  the  point,  such  as  we  are  obliged  to 
represent  it,  in  order  to  make  its  position  visible,  is 
in  reality  a  part  of  our  line,  and  our  points,  if  suffi- 
ciently multiplied  in  number  and  placed  side  by  side, 
would  compose  a  line  of  any  length  whatever.  But 
taking  the  mathematical  definition  of  a  point,  which 
denies  it  all  magnitude,  either  in  length,  breadth,  or 
thickness,  and  of  a  line,  which  is  asserted  to  possess 
length  only  without  breadth  or  thickness,  it  is  easy  to 
show  that  a  point  is  no  part  of  a  line,  by  making  it 
appear  that  the  shortest  line  can  be  cut  in  as  many 
points  as  the  longest,  which  may  be  done  in  the  fol- 
lowing manner.  Let  AB  be  any  straight  line,  from 


THE  DEFINITIONS  OF  GEOMETRY. 


195 


the  ends  of  which,  A  and  B,  draw  two  lines,  AF  and 
CB,  parallel  to  one  another.  Consider  AF  as  pro- 
duced without  limit,  and  in  CB  take  any  point  C,  from 
which  draw  lines  CE,  CF,  etc.,  to  different  points  in 
AF.  It  is  evident  that  for  each  point  E  in  AF  there 
is  a  distinct  point  in  AB,  viz.,  the  intersection  of  CE 
with  AB ; — for,  were  it  possible  that  two  points,  E 
and  F  in  AF,  could  be  thus  connected  with  the  same 
point  of  AB,  it  is  evident  that  two  straight  lines  would 
enclose  a  space,  viz.,  the  lines  CE  and  CF,  which 


C  B 

Fig.  x. 

both  pass  through  C,  and  would,  were  our  supposi- 
tion correct,  also  pass  through  the  same  point  in  AB. 
There  can  then  be  taken  as  many  points  in  the  finite 
or  unbounded  line  AB  as  in  the  indefinitely  extended 
line  AF. 

The  next  definition  which  we  shall  consider  is  that 
of  a  plane  surface.  The  word  plane  or  flat  is  as  hard 
to  define,  without  reference  to  any  thing  but  the  idea 
we  have  of  it,  as  it  is  easy  to  understand.  Neverthe- 
less the  practical  method  of  ascertaining  whether  or 
no  a  surface  is  plane,  will  furnish  a  definition,  not 


196  ON  THE  STUDY  OF  MATHEMATICS. 

such,  indeed,  #s  to  render  the  nature  of  a  plane  sur- 
face more  evident,  but  which  will  serve,  in  a  mathe- 
matical point  of  view,  as  a  basis  on  which  to  rest  the 
propositions  of  solid  geometry.  If  the  edge  of  a  ruler, 
known  to  be  perfectly  straight,  coincides  with  a  sur- 
face throughout  its  whole  length,  in  whatever  direc- 
tion it  may  be  placed  upon  that  surface,  we  conclude 
that  the  surface  is  plane.  Hence  the  definition  of  a 
plane  surface  is  that  in  which,  any  two  points  being 
taken,  the  straight  line  joining  these  points  lies  wholly 
upon  the  surface. 

Two  straight  lines  have  a  relation  to  one  another 
independent  altogether  of  their  length.  This  we  com- 
monly express  (for  among  the  most  common  ideas  are 
found  the  germ  of  every  geometrical  theory)  by  saying 
that  they  are  in  the  same  or  different  directions.  By 
the  direction  of  the  needle  we  ascertain  the  direction  in 
which  to  proceed  at  sea,  and  by  the  direction  in  which 
the  hands  of  a  clock  are  placed  we  tell  the  hour.  It 
remains  to  reduce  this  common  notion  to  a  more  pre- 
cise form. 

Suppose  a  straight  line  OA  to  be  given  in  magni- 
tude and  position,  and  to  remain  fixed  while  another 
line  OB,  at  first  coincident  with  OA,  is  made  to  move 
round  OA,  so  as  continually  to  vary  its  direction  with 
respect  to  OA.  The  process  of  opening  a  pair  of  com- 
passes will  furnish  an  illustration  of  this,  but  the  two 
lines  need  not  be  equal  to  one  another.  In  this  case 
the  opening  made  by  the  two  will  continually  increase, 


THE  DEFINITIONS  OF  GEOMETRY.  1 97 

and  this  opening  is  a  species  of  magnitude,  since  one 
opening  may  be  compared  with  another,  so  as  to  as- 
certain which  of  the  two  is  the  greater.  Thus  if  the 
figure  CPD  be  removed  from  its  place,  without  any 
other  change,  so  that  the  point  P  may  fall  on  O,  and 
the  line  PC  may  lie  upon  and  become  a  part  of  OA, 
or  OA  of  PC,  according  to  which  is  the  longer  of  the 
two,  then  if  the  opening  CPD  is  the  same  as  the  open- 
ing A  OB,  PD  will  lie  upon  OB  at  the  same  time  as 
PC  lies  upon  OA.  But  if  PD  does  not  then  lie  upon 
OB,  but  falls  between  OB  and  OA,  the  opening  CPD 


O  A  P  <- 

Fig.  2. 

is  less  than  the  opening  A  OB,  and  if  PD  does  not 
fall  between  OA  and  OB,  or  on  OB,  the  opening 
CPD  is  greater  than  the  opening  BOA.  To  this  spe- 
cies of  magnitude,  the  opening  of  two  lines,  the  name 
of  angle  is  given,  that  is  BO  is  said  to  make  an  angle 
with  OA.  The  difficulty  here  arises  from  this  magni- 
tude being  one,  the  measure  of  which  has  seldom  fal- 
len under  observation  of  those  who  begin  geometry. 
Every  one  has  measured  one  line  by  means  of  another, 
and  has  thus  made  a  number  the  representative  of  a 
length  ;  but  few,  at  this  period  of  their  studies,  have 


198  ON  THE  STUDY  OF  MATHEMATICS. 

been  accustomed  to  the  consideration,  that  one  open- 
ing may  be  contained  a  certain  number  of  times  in 
another,  or  may  be  a  certain  fraction  of  another. 
Nevertheless  we  may  find  measures  of  this  new  spe- 
cies of  magnitude  either  by  means  of  time,  length,  or 
number. 

One  magnitude  is  said  to  be  a  measure  of  another, 
when,  if  the  first  be  doubled,  trebled,  halved,  etc.,  the 
second  is  doubled,  trebled,  or  halved,  etc.;  that  is, 
when  any  fraction  or  multiple  of  the  first  corresponds 
to  the  same  fraction  or  multiple  of  the  second  in  the 
same  manner  as  the  first  does  to  the  second.  The  two 
quantities  need  not  be  of  the  same  kind  :  thus,  in  the 
barometer  the  height  of  the  mercury  (a  length)  meas- 
ures the  pressure  of  the  atmosphere  (a  weight)  ;  for  if 
the  barometer  which  yesterday  stood  at  28  inches,  to- 
day stands  at  29  inches,  in  which  case  the  height  of 
yesterday  is  increased  by  its  28th  part,  we  know  that 
the  atmospheric  pressure  of  yesterday  is  increased  by 
its  28th  part  to-day.  Again,  in  a  watch,  the  number 
of  hours  elapsed  since  twelve  o'clock  is  measured  by 
the  angle  which  a  hand  makes  with  the  position  it  oc- 
cupied at  twelve  o'clock.  In  the  spring  balances  a 
weight  is  measured  by  an  angle,  and  many  other  sim- 
ilar instances  might  be  given. 

This  being  premised,  suppose  a  line  which  moves 
round  another  as  just  described,  to  move  uniformly, 
that  is,  to  describe  equal  openings  or  angles  in  equal 
times.  Suppose  the  line  OA  to  move  completely 


THE  DEFINITIONS  OF  GEOMETRY. 


199 


round,  so  as  to  reassume  its  first  position  in  twenty- 
four  hours.  Then  in  twelve  hours  the  moving  line 
will  be  in  the  position  OB,  in  six  hours  it  will  be  in 
OC,  and  in  eighteen  hours  in  OD.  The  line  OC  is 
that  which  makes  equal  angles  with  OA  and  OB,  and 
is  said  to  be  at  right  angles,  or  perpendicular  to  OA 
and  OB.  Again,  OA  and  OB  which  are  in  the  same 

C 


D 

Fig.  3- 

right  line,  but  on  opposite  sides  of  the  point  O,  evi- 
dently make  an  opening  or  angle  which  is  equal  to 
the  sum  of  the  angles  AOC  and  COB,  or  equal  to  two 
right  angles.  A  line  may  also  be  said  to  make  with 
itself  an  opening  equal  to  four  right  angles,  since 
after  revolving  through  four  right  angles,  the  moving 
line  reassumes  its  original  position.  We  may  even 
carry  this  notion  farther :  for  if  the  moving  line  be  in 


200  ON  THE  STUDY  OF  MATHEMATICS. 

the  position  OE  when  P  hours  have  elapsed,  it  will 
recover  that  position  after  every  twenty-four  hours, 
that   is,  for   every   additional   four   right   angles  de- 
scribed; so  that  the  angle  AOE  is  equally  well  repre- 
sented by  any  of  the  following  angles : 
4  right  angles  -f  A  OE 
8  right  angles  -f  A  OE 
12  right  angles  -f  AOE,  etc. 

These  formulae  which  suppose  an  opening  greater 
than  any  apparent  opening,  and  which  take  in  and 
represent  the  fact  that  the  moving  line  has  attained 
its  position  for  the  second,  third,  fourth,  etc.,  time, 
since  the  commencement  of  the  motion,  are  not  of 
any  use  in  elementary  geometry ;  but  as  they  play  an 
important  part  in  the  application  of  algebra  to  the 
theory  of  angles,  we  have  thought  it  right  to  mention 
them  here. 

It  is  plain  also  that  we  may  conceive  the  line  OE 
to  make  two  openings  or  angles  with  the  original  po- 
sition OA  :  (1)  that  through  which  it  has  moved  to  re- 
cede from  OA  ;  (2)  that  through  which  it  must  move 
to  reach  OA  again.  The  first  (in  the  position  in  which 
we  have  placed  OA}  is  what  is  called  in  geometry  the 
angle  A  OE ;  the  second  is  more  simply  described  as 
composed  of  the  openings  or  angles  EOC,  COB, 
BOD,  DO  A,  and  is  not  used  except  in  the  application 
of  algebra  above  mentioned.*  Of  the  two  angles  just 

*But  use  is  made  of  it  in  some  modern  text-books  of  elementary  geome- 
try.-^. 


THE  DEFINITIONS  OF  GEOMETRY.  2OI 

alluded  to,  one  must  be  less  than  two  right  angles, 
and  the  second  greater ;  the  first  is  the  one  usually 
referred  to. 

It  is  plain  that  the  angle  or  opening  made  by  two 
lines  does  not  depend  upon  their  length  but  upon 
their  position ;  if  either  be  shortened  or  lengthened, 
the  angle  still  remains  the  same ;  and  if  while  the  an- 
gle increases  or  decreases  one  of  the  straight  lines 
containing  it  is  diminished,  the  angle  so  contained 
may  have  a  definite  and  given  magnitude  at  the  mo- 


Fig- 4 

ment  when  the  straight  line  disappears  altogether  and 
becomes  nothing.  For  example,  take  two  points  of 
any  curve  A£,  and  join  A  and  B  by  a  straight  line. 
Let  the  point  B  move  towards  A  \  it  is  evident  that 
the  angle  made  by  the  moving  line  with  AB  increases 
continually,  while  as  much  of  one  of  the  lines  contain- 
ing it  as  is  intercepted  by  the  curve,  diminishes  with- 
out limit.  When  this  intercepted  part  disappears  en- 
tirely, the  line  in  which  it  would  have  lain  had  it  had 
any  length,  has  reached  the  line  AG,  which  is  called 
the  tangent  of  the  curve. 


202  ON  THE  STUDY  OF  MATHEMATICS. 

In  elementary  geometry  two  equal  angles  lying  on 
different  sides  of  a  line,  such  as  A  OJS,  A  OH  (Fig.  3), 
would  be  considered  as  the  same.  In  the  application 
of  algebra,  they  would  be  considered  as  having  differ- 
ent signs,  for  reasons  stated  at  length  in  pages  112  et 
seq.,  of  the  first  part  of  this  Treatise.  It  is  also  com- 
mon in  the  latter  branch  of  the  science  to  measure 
angles  in  one  direction  only ;  for  example,  in  Figure  3 
the  angles  made  by  OE,  OF,  OG,  and  OH,  if  measured 
upwards  from  OAy  would  be  the  openings  through 
which  a  line  must  move  in  the  same  direction  from  OA, 
to  attain  those  positions;  and  the  second,  third,  and 
fourth  angles  would  be  greater  than  one,  two,  and 
three  right  angles  respectively. 

We  proceed  to  the  method  of  reasoning  in  geom- 
etry, or  rather  to  the  method  of  reasoning  in  general, 
since  there  is,  or  ought  to  be,  no  essential  difference 
between  the  manner  of  deducing  results  from  first 
principles,  in  any  science. 


CHAPTER  XIV. 

ON  GEOMETRICAL  REASONING. 

IT  is  evident  that  all  reasoning,  of  what  form  soever, 
can  be  reduced  at  last  to  a  number  of  simple  prop- 
ositions or  assertions ;  each  of  which,  if  it  be  not  self- 
evident,  depends  upon  those  which  have  preceded  it. 
Every  assertion  can  be  divided  into  three  distinct 
parts.  Thus  the  phrase,  "all  right  angles  are  equal," 
consists  of :  (1)  the  subject  spoken  of,  viz. ,  right  an- 
gles, which  is  here  spoken  of  universally,  since  every 
right  angle  is  a  part  of  the  subject ;  (2)  the  copula,  or 
manner  in  which  the  two  are  joined  together,  which 
is  generally  the  verb  is,  or  is  equal  to,  and  can  always 
be  reduced  to  one  or  the  other :  in  this  case  the  co- 
pula is  affirmative ;  (3)  the  predicate,  or  thing  asserted 
of  the  subject,  viz.,  equal  angles.  The  phrase,  thus 
divided,  stands  as  written  below  under  1,  and  is  called 
a  universal  affirmative.  The  second  is  called  a  particu- 
lar affirmative  proposition  ;  the  third  a  universal  nega- 
tive; the  fourth  a  particular  negative: 

1.  All  right  angles  are  equal  (magnitudes). 


204  ON  THE  STUDY  OF  MATHEMATICS. 

2.  Some  triangles  are  equilateral  (figures). 

3.  No  circle  is  convex  to  its  diameter. 

4.  Some  triangles  are  not  equilateral  (figures). 
Many  assertions  appear  in  a  form  which,  at  first 

sight,  cannot  be  reduced  to  one  of  the  preceding ;  the 
following  are  instances  of  the  change  which  it  is  nec- 
essary to  make  in  them  : 

1.  Parallel  lines  never  meet,  or  parallel  lines  are 
lines  which  never  meet. 

2.  The  angles  at  the  base  of  an  isosceles  triangle 
are  equal,  or  an  isosceles  triangle  is  a  triangle  having 
the  angles  at  the  base  equal. 

The  different  species  of  assertions,  and  the  argu- 
ments which  are  compounded  of  them,  may  be  dis- 
tinctly conceived  by  referring  them  all  to  one  species 
of  subject  and  predicate.  Since  every  assertion,  gen- 
erally speaking,  includes  a  number  of  individual  cases 
in  its  subject,  let  the  points  of  a  circle  be  the  subject 
and  those  of  a  triangle  the  predicate.  These  figures 
being  drawn,  the  four  species  of  assertions  just  alluded 
to  are  as  follows : 

1.  Every  point  of  the  circle  is  a  point  of  the  tri- 
angle, or  the  circle  is  contained  in  the  triangle. 

2.  Some  points  of  the  circle  are  points  of  the  tri- 
angle, or  part  of  the  circle  is  contained  in  the  tri- 
angle. 

3.  No  point  of  the  circle  is  a  point  of  the  triangle, 
or  the  circle  is  entirely  without  the  triangle. 


GEOMETRICAL  REASONING.  2O5 

4.  Some  points  of  the  circle  are  not  points  of  the 
triangle,  or  part  of  the  circle  is  outside  the  triangle. 

On  these  we  observe  that  the  second  follows  from 
the  first,  as  also  the  fourth  from  the  third,  since  that 
which  is  true  of  all  is  true  of  some  or  any ;  while  the 
first  and  third  do  not  follow  from  the  second  and 
fourth,  necessarily,  since  that  which  is  true  of  some 
only  need  not  be  true  of  all.  Again,  the  second  and 
fourth  are  not  necessarily  inconsistent  with  each  other 
for  the  same  reason.  Also  two  of  these  assertions 
must  be  true  and  the  others  untrue.  The  first  and 
the  third  are  called  contraries,  while  the  first  and 
fourth,  and  the  second  and  third  are  contradictory. 
The  converse  of  a  proposition  is  made  by  changing  the 
predicate  into  the  subject,  and  the  subject  into  the 
predicate.  No  mistake  is  more  common  than  con- 
founding together  a  proposition  and  its  converse,  the 
tendency  to  which  is  rather  increased  in  those  who 
begin  geometry,  by  the  number  of  propositions  which 
they  find,  the  converses  of  which  are  true.  Thus  all 
the  definitions  are  necessarily  conversely  true,  since 
the  identity  of  the  subject  and  predicate  is  not  merely 
asserted,  but  the  subject  is  declared  to  be  a  name 
given  to  all  those  magnitudes  which  have  the  proper- 
ties laid  down  in  the  predicate,  and  to  no  others. 
Thus  a  square  is  a  four-sided  figure  having  equal 
sides  and  one  right  angle,  that  is,  let  every  four-sided 
figure  having,  etc.,  be  called  a  square,  and  let  no  other 
figure  be  called  by  that  name,  whence  the  truth  of  the 


206  ON  THE  STUDY  OF  MATHEMATICS. 

converse  is  evident.  Also  many  of  the  facts  proved 
in  geometry  are  conversely  true.  Thus  all  equilateral 
triangles  are  equiangular,  from  which  it  is  proved  that 
all  equiangular  triangles  are  equilateral.  Of  the  first 
species  of  assertion,  the  universal  affirmative,  the  con- 
verse is  not  necessarily  true.  Thus  "  every  point  in 
figure  A  is  a  point  of  B,"  does  not  imply  that  "every 
point  of  B  is  a  point  of  A,"  although  this  may  be  the 
case,  and  is,  if  the  two  figures  coincide  entirely.  The 
second  species,  the  particular  affirmative,  is  conversely 
true,  since  if  some  points  of  A  are  points  of  B,  some 
points  of  B  are  also  points  of  A.  The  first  species  of 
assertion  is  conversely  true,  if  the  converse  be  made 
to  take  the  form  of  the  second  species  :  thus  from 
"all  right  angles  are  equal/'  it  may  be  inferred  that 
"seme  equal  magnitudes  are  right  angles."  The  third 
species,  the  universal  negative,  is  conversely  true, 
since  if  "no  point  of  B  is  a  point  of  A,"  it  may  be  in- 
ferred that  "no  point  of  A  is  a  point  of  B."  The 
fourth  species,  the  particular  negative,  is  not  neces- 
sarily conversely  true.  From  "some  points  of  A  are 
not  points  of  B"  or  A  is  not  entirely  contained  within 
B,  we  can  infer  nothing  as  to  whether  B  is  or  is  not 
entirely  contained  in  A.  It  is  plain  that  the  converse 
of  a  proposition  is  not  necessarily  true,  if  it  says  more 
either  of  the  subject  or  predicate  than  was  said  before. 
Now  "every  equilateral  triangle  is  equiangular,"  does 
not  speak  of  all  equiangular  triangles,  but  asserts  that 
among  all  possible  equiangular  triangles  are  to  be 


GEOMETRICAL  REASONING. 

found  all  the  equilateral  ones.  There  may  then,  for 
anything  to  the  contrary  to  be  discovered  in  our  as- 
sertion, be  classes  of  equiangular  triangles  not  in- 
cluded under  this  assertion,  of  which  we  can  therefore 
say  nothing.  But  in  saying  "no  right  angles  are  un- 
equal," that  which  we  exclude,  we  exclude  from  all 
unequal  angles,  and  therefore  "no  unequal  angles  are 
right  angles  "  is  not  more  general  than  the  first. 

The  various  assertions  brought  forward  in  a  geo- 
metrical demonstration  must  be  derived  in  one  of  the 
following  ways : 

I.  From  definition.  This  is  merely  substituting, 
instead  of  a  description,  the  name  which  it  has  been 
agreed  to  give  to  whatever  bears  that  description.  No 
definition  ought  to  be  introduced  until  it  is  certain 
that  the  thing  defined  is  really  possible.  Thus  though 
parallel  lines  are  defined  to  be  "lines  which  are  in 
the  same  plane,  and  which  being  ever  so  far  produced 
never  meet,"  the  mere  agreement  to  call  such  lines, 
should  they  exist,  by  the  name  of  parallels,  is  no  suffi- 
cient ground  to  assume  that  they  do  exist.  The  defi- 
nition is  therefore  inadmissible  until  it  is  really  shown 
that  there  are  such  things  as  lines  which  being  in  the 
same  plane  never  meet.  Again,  before  applying  the 
name,  care  must  be  taken  that  all  the  circumstances 
connected  with  the  definition  have  been  attended  to. 
Thus,  though  in  plane  geometry,  where  all  lines  are 
in  one  plane,  it  is  sufficient  that  two  lines  would  never 
meet  though  ever  so  far  produced,  to  call  them  par- 


208  ON  THE  STUDY  OF  MATHEMATICS. 

allel,  yet  in  solid  geometry  the  first  circumstance  must 
be  attended  to,  and  it  must  be  shown  that  lines  are  in 
the  same  plane  before  the  name  can  be  applied.  Some 
of  the  axioms  come  so  near  to  definitions  in  their  na- 
ture, that  their  place  may  be  considered  as  doubtful. 
Such  are,  "  the  whole  is  greater  than  its  part,"  and 
"magnitudes  which  entirely  coincide  are  equal  to  one 
another." 

II.  From  hypothesis.  In  the  statement  of  every 
proposition,  certain  connexions  are  supposed  to  exist 
from  which  it  is  asserted  that  certain  consequences 
will  follow.  Thus  "in  an  isosceles  triangle  the  angles 
at  the  base  are  equal,"  or,  "  if  a  triangle  be  isosceles 
the  angles  at  the  base  will  be  equal."  Here  the  hy- 
pothesis or  supposition  is  that  the  triangle  has  two 
equal  sides,  the  consequence  asserted  is  that  the  an- 
gles at  the  base  or  third  side  will  be  equal.  The  con- 
sequence being  only  asserted  to  be  true  when  the  an- 
gle is  isosceles,  such  a  triangle  is  supposed  to  be  taken 
as  the  basis  of  the  reasonings,  and  the  condition  that 
its  two  sides  are  equal,  when  introduced  in  the  proof, 
is  said  to  be  introduced  by  hypothesis. 

In  order  to  establish  the  result  it  may  be  necessary 
to  draw  other  lines,  etc.,  which  are  not  mentioned  in 
the  first  hypothesis.  These,  when  introduced,  form 
what  is  called  the  construction. 

There  is  another  species  of  hypothesis  much  in 
use,  principally  when  it  is  required  to  deduce  the  con- 
verse of  a  theorem  from  the  theorem  itself.  Instead 


GEOMETRICAL  REASONING.  2OQ 

of  proving  the  consequence  directly,  the  contradictory 
of  the  consequence  is  assumed  to  hold  good,  and  if 
from  this  new  hypothesis,  supposed  to  exist  together 
with  the  old  one,  any  evidently  absurd  result  can  be 
derived,  such  as  that  the  whole  is  greater  than  its 
part,  this  shows  that  the  two  hypotheses  are  not  con- 
sistent, and  that  if  the  first  be  true,  the  second  cannot 
be  so.  But  if  the  second  be  not  true,  its  contradic- 
tory is  true,  which  is  what  was  required  to  be  proved. 
III.  From  the  evidence  of  the  assertions  themselves. 
The  propositions  thus  introduced  without  proof  are 
only  such  as  are  in  their  nature  too  simple  to  ad- 
mit of  it.  They  are  called  axioms.  But  it  is  neces- 
sary to  observe,  that  the  claim  of  an  assertion  to  be 
called  an  axiom  does  not  depend  only  on  its  being 
self-evident.  Were  this  the  case  many  propositions 
which  are  always  proved  might  be  assumed  ;  for  ex- 
ample, that  two  sides  of  a  triangle  are  greater  than 
the  third,  or  that  a  straight  line  is  the  shortest  dis- 
tance between  two  points.  In  addition  to  being  self- 
evident,  it  must  be  incapable  of  proof  by  any  other 
means,  and  it  is  one  of  the  objects  of  geometry  to  re- 
duce the  demonstrations  to  the  least  possible  number 
of  axioms.  There  are  only  two  axioms  which  are  dis- 
tinctly geometrical  in  their  nature,  viz.,  "two  straight 
lines  cannot  enclose  a  space, "  and  "  through  each 
point  outside  a  line,  not  more  than  one  parallel  to 
that  line  can  be  drawn."  All  the  rest  of  the  proposi- 
tions commonly  given  as  axioms  are  either  arithmet- 


210  ON  THE  STUDY  OF  MATHEMATICS. 

ical  in  their  nature;  such  as  "the  whole  is  greater 
than  its  part,"  "the  doubles  of  equals  are  equals," 
etc.;  or  mere  definitions,  such  as  "magnitudes  which 
entirely  coincide  are  equal  "j  or  theorems  admitting 
of  proof,  such  as  "all  right  angles  are  equal."  There 
is  however  one  more  species  of  self-evident  proposi- 
tion, the  postulate  or  self-evident  problem,  such  as 
the  possibility  of  drawing  a  right  line,  etc. 

IV.  From  proof  already  given.  What  has  been 
proved  once  may  be  always  taken  for  granted  after- 
wards. It  is  evident  that  this  is  merely  for  the  sake 
of  brevity,  since  it  would  be  possible  to  begin  from 
the  axioms  and  proceed  direct  to  the  proof  of  any  one 
proposition,  however  far  removed  from  them ;  and 
this  is  an  exercise  which  we  recommend  to  the  stu- 
dent. Thus  much  for  the  legitimate  use  of  any  single 
assertion  or  proposition.  We  proceed  to  the  manner 
of  deducing  a  third  proposition  from  two  others. 

It  is  evident  that  no  assertion  can  be  the  direct 
and  necessary  consequence  of  two  others,  unless  those 
two  contain  something  in  common,  or  which  is  spoken 
of  in  both.  In  many,  nay  most,  cases  of  ordinary  con- 
versation and  writing,  we  leave  out  one  of  the  asser- 
tions, which  is,  usually  speaking,  very  evident,  and 
make  the  other  assertion  followed  by  the  consequence 
of  both.  Thus,  "Geometry  is  useful,  and  therefore 
ought  to  be  studied,"  contains  not  only  what  is  ex- 
pressed, but  also  the  following,  "That  which  is  useful 
ought  to  be  studied ; "  for  were  this  not  admitted,  the 


GEOMETRICAL  REASONING.  2ll 

former  assertion  would  not  be  necessarily  true.     This 
may  be  written  thus  : 

Every  thing  useful  is  what  ought  to  be  stud- 
ied. 
Geometry  is   useful,   therefore   geometry  is 

what  ought  to  be  studied. 

This,  in  its  present  state,  is  called  a  syllogism,  and 
may  be  compared  with  the  following,  from  which  it 
only  differs  in  the  things  spoken  of,  and  not  in  the 
manner  in  which  they  are  spoken  of. 

Every  point  of  the  circle  is  a  point  of  the  tri- 
angle. 

The  point  B  is  a  point  of  the  circle. 
Therefore  the  point  B  is  a  point  of  the  tri- 
angle. 

Here  a  connexion  is  established  between  the  point 
B  and  the  points  of  the  triangle  (viz.,  that  the  first  is 
one  of  the  second)  by  comparing  them  with  the  points 
of  the  circle ;  that  which  is  asserted  of  every  point  of 
the  circle  in  the  first  can  be  asserted  of  the  point  B, 
because  from  the  second  B  is  one  of  these  points. 
Again,  in  the  former  argument,  whatever  is  asserted 
of  every  thing  useful  is  true  of  geometry,  because  ge- 
ometry is  useful. 

The  common  term  of  the  two  propositions  is  called 
the  middle  term,  while  the  predicate  and  subject  of  the 
conclusion  are  called  the  major  and  minor  terms,  re- 
spectively. The  two  first  assertions  are  called  the 
major  and  minor  premisses,  and  the  last  the  conclusion. 


212  ON  THE  STUDY  OF  MATHEMATICS. 

Suppose  now  the  two  premisses  and  conclusion  of  the 
syllogism  just  quoted  to  be  varied  in  every  possible 
way  from  affirmative  to  negative,  from  universal  to 
particular,  and  vice  versa,  where  the  number  of  changes 
will  be  4x^X4,  or  64  (called  moods);  since  each 
proposition  may  receive  four  different  forms,  and  each 
form  of  one  may  be  compounded  with  any  of  the  other 
two.  And  these  may  be  still  further  varied,  if  instead 
of  the  middle  term  being  the  subject  of  the  first,  and 
the  predicate  of  the  second,  this  order  be  reversed,  or 
if  the  middle  term  be  the  subject  of  both,  or  the  pred- 
icate of  both,  which  will  give  four  different  figures,  as 
they  are  called,  to  each  of  the  sixty-four  moods  above 
mentioned.  But  of  these  very  few  are  correct  deduc- 
tions, and  without  entering  into  every  case  we  will 
state  some  general  rules,  being  the  methods  which 
common  reason  would  take  to  ascertain  the  truth  or 
falsehood  of  any  one  of  them,  collected  and  general- 
ised.* 

I.   The  middle   term  must  be  the  same  in  both 


*Whately's  Logic,  page  76,  third  edition.  A  work  which  should  be  read 
by  all  mathematical  students.  [Whately's  Logic  is  procurable  in  modern  edi- 
tions, many  of  which  were,  until  recently,  widely  read  in  our  academies  and 
colleges.  The  following  works  in  which  the  same  material  is  presented  in  a 
shape  more  comforming  to  modern  methods  may  be  mentioned :  T.  Fowler's 
Elements  of  Deductive  Logic ;  Bain's  Logic;  Venn's  Empirical  Logic  and  Sym- 
bolical Logic;  Keynes's  Formal  Logic;  Carveth  Read's  Logic,  Deductive  and 
Inductive;  Mill's  System  of  Logic  (a.  discussion  rather  than  a  presentation). 
Strictly  contemporary  logic  will  be  found  represented  in  the  following  works 
in  English:  Jevons's  Principles  of  Science  and  Studies  in  Deductive  Logic; 
Bradley's  Principles  of  Logic ;  Sidgwick's  Process  of  Argument;  Bosanquet's 
Logic:  or,  the  Morphology  of  Knowledge;  and  the  same  author's  Essentials  of 
Logic:  Sigwart's  Logic,  recently  translated  from  the  German;  and  Ueber- 
weg's  System  of  Logic  and  History  of  Logical  Doctrines. — Ed.] 


GEOMETRICAL  REASONING.  213 

premisses,  by  what  has  just  been  observed ;  since  in 
the  comparison  of  two  things  with  one  and  the  same 
third  thing,  in  order  to  ascertain  their  connexion  or 
discrepancy,  consists  the  whole  of  reasoning.  Thus, 
the  deduction  without  further  process  of  the  equation 
a2-}-^  =  f2  from  the  proposition,  which  proves  that 
the  sum  of  the  squares  described  on  the  sides  of  a 
right-angled  triangle  is  equal  to  the  square  on  its  hy- 
pothenuse,  a,  b,  and  c  being  the  number  of  linear  units 
in  the  sides  and  the  hypothenuse,  is  incorrect,  since 
syllogistically  stated  the  argument  would  stand  thus : 
The  sum  of  the  squares  of  the 

lines  a  and  6 are          j          tities> 

and 

the  square  of  the  line  c 

#2  +  P  }          [  the  sum  of  the  squares  of  a  and  £, 
and      \  are  \  and 

f2 J          I  the  square  of  c. 

a2  +  #M 

Therefore          and     \  are  equal  quantities. 
S J 

Here  the  term  square  in  the  major  premiss  has  its 
geometrical,  and  in  the  minor  its  algebraical  sense, 
being  in  the  first  a  geometrical  figure,  and  in  the  sec- 
ond an  arithmetical  operation.  The  term  of  compari- 
son is  not  therefore  the  same  in  both,  and  the  conclu- 
sion does  not  therefore  follow  from  the  premisses. 

The  same  error  is  committed  if  all  that  can  be  con- 
tained under  the  middle  term  be  not  spoken  of  either 
in  the  major  or  minor  premiss.  For  if  each  premiss 


214  ON  THE  STUDY  OF  MATHEMATICS. 

mentions  only  a  part  of  the  middle  term,  these  parts 
may  be  different,  and  the  term  of  comparison  really 
different  in  the  two,  though  passing  under  the  same 
name  in  both.  Thus, 

All  the  triangle  is  in  the  circle, 

All  the  square  is  in  the  circle, 

proves  nothing,  since  the  square  may,  consistently 
with  these  conditions,  be  either  wholly,  partly,  or  not 
at  all  contained  in  the  triangle.  In  fact,  as  we  have 
before  shown,  each  of  these  assertions  speaks  of  a  part 
of  the  circle  only.  The  following  is  of  the  same  kind  : 

Some  of  the  triangle  is  in  the  circle. 

Some  of  the  circle  is  not  in  the  square,  etc. 

II.  If  both  premisses  are  negative,  no  conclusion 
can  be  drawn.   For  it  can  evidently  be  no  proof  either 
of  agreement  or  disagreement  that  two  things  both 
disagree  with  a  third.     Thus  the  following  is  incon- 
clusive : 

None  of  the  circle  is  in  the  triangle. 
None  of  the  square  is  in  the  circle. 

III.  If  both  premisses  are  particular,  no  conclusion 
can  be  drawn,  as  will  appear  from  every  instance  that 
can  be  taken,  thus : 

Some  of  the  circle  is  in  the  triangle. 
Some  of  the  square  is  not  in  the  circle, 
proves  nothing. 

IV.  In  forming  a  conclusion,  where  a  conclusion 
can  be  formed,  nothing  must  be  asserted  more  gener- 


GEOMETRICAL  REASONING.  215 

ally  in  the  conclusion  than  in  the  premisses.  Thus,  if 
from  the  following, 

All  the  triangle  is  in  the  circle, 

All  the  circle  is  in  the  square, 

we  would  draw  a  conclusion  in  which  the  square 
should  be  the  subject,  since  the  whole  square  is  not 
mentioned  in  the  minor  premiss,  but  only  part  of  it, 
the  conclusion  must  be, 

Part  of  the  square  is  in  the  triangle. 

V.  If  either  of  the  premisses  be  negative,  the  con- 
clusion must  be  negative.    For  as  both  premisses  can- 
not be  negative,  there  is  asserted  in  one  premiss  an 
agreement  between  the  term  of  the  conclusion  and 
the  middle  term,  and  in  the  other  premiss  a  disagree- 
ment between  the  other  term  of  the  conclusion,  and 
the  same  middle  term.     From  these  nothing  can  be 
inferred  but  a  disagreement  or  negative  conclusion. 
Thus,  from 

None  of  the  circle  is  in  the  triangle, 
All  the  circle  is  in  the  square, 
can  only  be  inferred, 

Some  of  the  square  is  not  in  the  triangle. 

VI.  If  either  premiss  be  particular,  the  conclusion 
must  be  particular.     For  example,  from 

None  of  the  circle  is  in  the  triangle, 
Some  of  the  circle  is  in  the  square, 
we  deduce, 

Some  of  the  square  is  not  in  the  triangle. 
If  the  student  now  applies  these  rules,  he  will  find 


21 6  ON  THE  STUDY  OF  MATHEMATICS. 

that  of  the  sixty-four  moods  eleven  only  are  admis- 
sible in  any  case ;  and  in  applying  these  eleven  moods 
to  the  different  figures  he  will  also  find  that  some  of 
them  are  not  admissible  in  every  figure,  and  some  not 
necessary,  on  account  of  the  conclusion,  though  true, 
not  being  as  general  as  from  the  premisses  it  might  be. 
This  he  may  do  either  by  reasoning  or  by  actual  in- 
spection of  the  figures,  drawn  and  arranged  according 
to  the  premisses.  The  admissible  moods  are  nineteen 
in  number,  and  are  as  follows,  where  A  at  the  begin- 
ning of  a  proposition  signifies  that  it  is  a  universal 
affirmative,  E  a  universal  negative,  /  a  particular 
affirmative,  O  a  particular  negative. 

Figure  I.  The  middle  term  is  the  subject  of  the 
major,  and  the  predicate  of  the  minor  premiss. 

1.*  A  All  the  O  is  in  the  A 

A  All  the  n  is  in  the  O 

.  • .    A  All  the  n  is  in  the  A 

2.  E  None  of  the  O  is  in  the  A 
A  All  the  n  is  in  the  Q 

.  • .    E     None  of  the    n  is  in  the  A 

3.  A     All  the  O  is  in  the  A 
/      Some  of  the  n  is  in  the  O 

.  *.    /      Some  of  the  n  is  in  the  A 

4.  E     None  of  the  O  is  in  the  A 
/      Some  of  the  n  is  in  the  O 

.•.    O     Some  of  the  n  is  not  in  A 

*This,  and  3,  are  the  most  simple  of  all  the  combinations,  and  the  most 
frequently  used,  especially  in  geometry. 


GEOMETRICAL  REASONING.  217 

Figure  II.   The  middle  term  is  the  predicate  of 
both  premisses. 

1.  E  None  of  the   A  is  in  the  O 
A  All  the  n  is  in  the  Q 

.  •.    E  None  of  the   n  is  in  the  A 

2.  A  All  the  A  is  in  the  Q 
E  None  of  the    n  is  in  the  O 

.  • .    E  None  of  the    n  is  in  the  A 

3.  E  None  of  the   A  is  in  the  Q 
/  Some  of  the  n  is  in  the  O 

.  •.    O  Some  of  the  n  is  not  in  A 

4.  A  All  the  A  is  in  the  O 

0  Some  of  the  n  is  not  in  O 
.  • .    O  Some  of  the  D  is  not  in  A 

Figure  III.  The  middle  term  is  the  subject  of  both 
premisses. 

1.  A  All  the  O  is  in  the  A 
A  All  the           O  is  in  the  n 

.  • .    7  Some  of  the   D  is  in  the  A 

2.  /  Some  of  the  O  is  in  the  A 
A  All  the  O  is  in  the  n 

.  • .    /  Some  of  the  n  is  in  the  A 

3.  A  All  the  O  is  in  the  A 

1  Some  of  the  O  is  in  the  n 
.  • .    /  Some  of  the  n  is  in  the  A 

4.  E  None  of  the   O  is  in  the  A 
A  All  the  O  is  in  the  n 

.  • .    O  Some  of  the   n  is  not  in  A 

5.  O  Some  of  the  O  is  not  in  A 


2l8  ON  THE  STUDY  OF  MATHEMATICS. 

A     All  the  O  is  in  the  n 

.  • .    O     Some  of  the  n  is  not  in  A 
6.     E     None  of  the   O  is  in  the  A 
/      Some  of  the  O  is  in  the  n 
.  • .    O     Some  of  the  D  is  not  in  A 
Figure  IV.  The  middle  term  is  the  predicate  of 
the  major,  and  the  subject  of  the  minor  premiss. 

1.  A     All  the  A  is  in  the  Q 
A      All  the            O  is  in  the  n 

.  • .   /      Some  of  the  n  is  in  the  A 

2.  A     All  the  A  is  in  the  O 
E     None  of  the   O  is  in  the  n 

.  • .   E     None  of  the   n  is  in  the  A 

3.  /      Some  of  the  A  is  in  the  O 
A     All  the  O  is  in  the  n 

.  • .    /      Some  of  the   n  is  in  the  A 

4.  E     None  of  the   A  is  in  the  O 
A     All  the  O  is  in  the  n 

.  • .    O     Some  of  the  n  is  not  in  A 

5.  E     None  of  the   A  is  in  the  O 
/      Some  of  the  O  is  in  the  n 

.  • .    O     Some  of  the  n  is  not  in  A 
We  may  observe  that  it  is  sometimes  possible  to 
condense  two  or  more  syllogisms  into  one  argument, 
thus  :  Every  A  is  B  (1), 

Every  B  is  C  (2), 
Every  C  is  D  (3), 
Every  D  is  E  (4), 
Therefore  Every  A  is  E  (5), 


GEOMETRICAL  REASONING.  2ig 

is  equivalent  to  three  distinct  syllogisms  of  the  form 
Fig.  l.j  these  syllogisms  at  length  being  (1),  (2),  a\ 
a,  (3),  b;  b,  (4),  (5). 

The  student,  when  he  has  well  considered  each  of 
these,  and  satisfied  himself,  first  by  the  rules,  and 
afterwards  by  inspection,  that  each  of  them  is  legiti- 
mate ;  and  also  that  all  other  moods,  not  contained 
in  the  above,  are  not  allowable,  or  at  least  do  not  give 
the  most  general  conclusion,  should  form  for  himself 
examples  of  each  case,  for  instance  of  Fig.  Ill,  3 : 

The  axioms  constitute  part  of  the  basis  of 

geometry. 

Some  of  the  axioms  are  grounded  on  the  evi- 
dence of  the  senses. 
.  • .  Some  evidence  derived  from  the  senses  is 

part  of  the  basis  of  geometry. 

He  should  also  exercise  himself  in  the  first  princi- 
ples of  reasoning  by  reducing  arguments  as  found  in 
books  to  the  syllogistic  form.  Any  controversial  or 
argumentative  work  will  furnish  him  with  a  sufficient 
number  of  instances. 

Inductive  reasoning  is  that  in  which  a  universal 
proposition  is  proved  by  proving  separately  every  one 
of  its  particular  cases.  As  where,  for  example,  a 
figure,  A  BCD,  is  proved  to  be  a  rectangle  by  proving 
each  of  its  angles  separately  to  be  a  right  angle,  or 
proving  all  the  premisses  of  the  following,  from  which 
the  conclusion  follows  necessarily : 


22O  ON  THE  STUDY  OF  MATHEMATICS. 

The  angles  at  A,  B,  C,  and  D  are  all  the  an- 
gles of  the  figure  A  BCD. 
A  is  a  right  angle, 
B  is  a  right  angle, 
C  is  a  right  angle, 
Z?  is  a  right  angle, 
Therefore  all  the  angles  of  the  figure  A  BCD 

are  right  angles. 

This  may  be  considered  as  one  syllogism  of  which 
the  minor  premiss  is, 

A,  B,  C,  and  D  are  right  angles, 
where  each  part  is  to  be  separately  proved. 

Reasoning  a  fortiori,  is  that  contained  in  Fig.  I.  1. 
in  a  different  form,  thus  :  A  is  greater  than  B,  B  is 
greater  than  C ;  a  fortiori  A  is  greater  than  C ;  which 
may  be  also  stated  as  follows  : 

The  whole  of  B  is  contained  in  A, 
The  whole  of  C  is  contained  in  By 
Therefore  C  is  contained  in  A. 
The  premisses  of  the  second  do  not  necessarily  im- 
ply as  much  as  those  of  the  first ;  the  complete  reduc- 
tion we  leave  to  the  student. 

The  elements  of  geometry  present  a  collection  of 
such  reasonings  as  we  have  just  described,  though  in 
a  more  condensed  form.  It  is  true  that,  for  the  con- 
venience of  the  learner,  it  is  broken  up  into  distinct 
propositions,  as  a  journey  is  divided  into  stages  ;  but 
nevertheless,  from  the  very  commencement,  there  is 
nothing  which  is  not  of  the  nature  just  described.  We 


GEOMETRICAL  REASONING. 


221 


present  the  following  as  a  specimen  of  a  geometrical 
proposition  reduced  nearly  to  a  syllogistic  form.  To 
avoid  multiplying  petty  syllogisms,  we  have  omitted 
some  few  which  the  student  can  easily  supply. 

Hypothesis.  —  ABC  is  a  right-angled  triangle  the 
right  angle  being  at  A. 

Consequence.  —  The  squares  on  AB  and  -^Care  to- 
gether equal  to  the  square  on  BC. 

Construction-. 


and  BA  describe  squares, 
produce  DB  to  meet  EFt 
produced,  if  necessary,  in 
G,  and  through  A  draw 
HAK  parallel  to  BD. 

Demonstration. 

I.  Conterminous  sides 
of  a  square  are  at  right 
angles    to    one    another. 
(Definition.) 

EB  and  BA  are  conter- 
minous sides  of  a  square. 
(Construction.) 

.  •  .  EB  and  BA  are  at  right  angles. 

II.  A  similar  syllogism  to  prove  that  DB  and  BC 
are  at  right  angles,  and  another  to  prove  that  GB  and 
BC  are  at  right  angles. 

III.  Two  right  lines  drawn  perpendicular  to  two 
other  right  lines  make  the  same  angle  as  those  others 


222  ON  THE  STUDY  OF  MATHEMATICS. 

(already  proved)  ;  EB  and  BG  and  AB  and  BC  are 
two  right  lines,  etc.,  (I.  II.). 

.  • .  The  angle  EBG  is  equal  to  ABC. 

IV.  All  sides  of  a  square  are  equal.     (Definition.) 
AB  and  BE  are  sides  of  a  square.    (Construction.) 
.  • .  AB  and  BE  are  equal. 

V.  All  right  angles  are  equal.     (Already  proved.) 
BEG  and  BAG  are  right  angles.   (Hypothesis  and 

construction.) 

.  • .  BEG  and  BAC  are  equal  angles. 

VI.  Two  triangles  having  two  angles  of  one  equal 
to  two  angles  of  the  other,  and  the  interjacent  sides 
equal,  are  equal  in  all  respects.     (Proved.) 

BEG  and  BAC  are  two  triangles  having  BEG  and 
EBG  respectively  equal  to  BAC  and  ABC  and  the 
sides  EB  and  BA  equal.  (III.  IV.  V.) 

.-.  The  triangles  BEG,  BAC  are  equal  in  all  re- 
spects. 

VII.  BG  is  equal  to  BC.     (VI.) 

BC  is  equal  to  BD.     (Proved  as  IV.) 
.-.  BGis  equal  to  BD. 

VIII.  A  four-sided   figure  whose   opposite   sides 
are  parallel  is  a  parallelogram.    (Definition.)    BGHA 
and  BPKD  are  four-sided  figures,   etc,     (Construc- 
tion.) 

.  • .  BGHA  and  BPKD  are  parallelograms. 

IX.  Parallelograms  upon  the  same  base  and  be- 
tween the  same  parallels  are  equal.   (Proved.)   EBAF 
and  BGHA,  are  parallelograms,  etc.    (Construction.) 


GEOMETRICAL  REASONING.  223 

.  • .  EBAFand.  BGHA  are  equal. 

X.  Parallelograms  on  equal  bases  and  between  the 
same  parallels,  are  equal.     (Proved.) 

BGHA  and  BDKP  are  parallelograms,  etc.  (Con- 
struction.) 

.  • .  BGHA  and  BDKP  are  equal. 

XI.  EBAF'is  equal  to  BGHA.     (IX.) 
BGHA  is  equal  to  BDKP.     (X.) 

.  • .  EBAF  (that  is  the  square  on  AB)  is  equal  to 
BDKP. 

XII.  A  similar  argument  from  the  commencement 
to  prove  that  the  square  on  A  C  is  equal  to  the  rectan- 
gle CPK. 

XIII.  The  rectangles  BK  and  CK  are  together 
equal  to  the  square  on  BC.     (Self-evident  from  the 
construction.) 

The  squares  on  BA  and  A  C  are  together  equal  to 
the  rectangles  BK  and  CK.  (Self-evident  from  XI 
and  XII.) 

.  • .  The  squares  on  BA  and  A  C  are  together  equal 
to  the  square  on  BC. 

Such  is  an  outline  of  the  process,  every  step  of 
which  the  student  must  pass  through  before  he  has 
understood  the  demonstration.  Many  of  these  steps 
are  not  contained  in  the  book,  because  the  most  ordi- 
nary intelligence  is  sufficient  to  suggest  them,  but  the 
least  is  as  necessary  to  the  process  as  the  greatest. 
Instead  of  writing  the  propositions  at  this  length,  the 


224 


ON  THE  STUDY  OF  MATHEMATICS. 


student  is  recommended  to  adopt  the  plan  which  we 
now  lay  before  him. 

Hyp.  1  ABC  is  a  triangle,  right-an- 

gled at  A. 

Constr.  2  a  On  BA  describe  a  square 

SAFE. 

3  a  On  BC  describe  a  square. 

4  Produce  BD  to  meet  EF,  pro- 

duced if  necessary,  in  G. 

5  b  Through  A  draw  HAK  par- 

allel to  BD. 
Demonst.    6      2,  Def.      EBA  is  a  right  angle. 

7  3  £.#C  is  a  right  angle. 

8  6,  7,  c     LEBG  is  equal  to  /_ABC. 

9  2,  1,  </     /_BEG  is  equal  to  /_BAC. 

10  2  jET?  is  equal  to  AB. 

11  8,  9,  10,  e   The  triangles  .#££  and  ABC 

are  equal. 

12  11,  3       BG  is  equal  to  BD. 

13  5,  2,  Def.    AHGB  is  a  parallelogram. 

14  5,  3,  Def.    BPDK\s  a  parallelogram. 

15  13,  2,  y    AHGB  and  ABEF  are  equal. 

16  13,14,^    ^ZT£#  and  ^Z^ST  are  equal. 

17  15,  16       BPDK  and  the  square  on 

are  equal. 

f       By 
18{    similar 


and  the  S1uare  on 


GEOMETRICAL  REASONING.  225 

19       17,  18      The  square  on  BC  is  equal  to 
the  squares  on  BA  and  A  C. 
a,  b     Here  refer  to  the  necessary  problems. 
c       If  two  lines  be  drawn  at  right  angles  to 
two  others,  the  angles  made  by  the 
first  and  second  pair  are  equal. 
d       All  right  angles  are  equal. 
e       Two  triangles  which  have  two  angles  of 
one  equal  to  two  angles  of  the  other, 
and  the  interjacent  sides  equal,  are 
equal  in  all  respects. 

/,  g  Parallelograms  on  the  same  or  equal 
bases,  and  between  the  same  paral- 
lels, are  equal. 

The  explanation  of  this  is  as  follows :  the  whole 
proposition  is  divided  into  distinct  assertions,  which 
are  placed  in  separate  consecutive  paragraphs,  which 
paragraphs  are  numbered  in  the  first  column  on  the 
left  ;  in  the  second  column  on  the  left  we  state  the 
reasons  for  each  paragraph,  either  by  referring  to  the 
preceding  paragraphs  from  which  they  follow,  or  the 
preceding  propositions  in  which  they  have  been 
proved.  In  the  latter  case  a  letter  is  placed  in  the 
column,  and  at  the  end,  the  enunciation  of  the  propo- 
sition there  used  is  written  opposite  to  the  letter.  By 
this  method,  the  proposition  is  much  shortened,  its 
more  prominent  parts  are  brought  immediately  under 
notice,  and  the  beginner,  if  he  recollect  the  preceding 
propositions  perfectly  well,  is  not  troubled  by  the 


226  ON  THE  STUDY  OF  MATHEMATICS. 

repetition  of  prolix  enunciations,  while  in  the  contrary 
case  he  has  them  at  hand  for  reference. 

In  all  that  has  been  said,  we  have  taken  instances 
only  of  direct  reasoning,  that  is,  where  the  required 
result  is  immediately  obtained  without  any  reference 
to  what  might  have  happened  if  the  result  to  be  proved 
had  not  been  true.  But  there  are  many  propositions 
in  which  the  only  possible  result  is  one  of  two  things 
which  cannot  be  true  at  the  same  time,  and  it  is  more 
easy  to  show  that  one  is  not  the  truth,  than  that  the 
other  is.  This  is  called  indirect  reasoning ;  not  that 
it  is  less  satisfactory  than  the  first  species,  but  be- 
cause, as  its  name  imports,  the  method  does  not  ap- 
pear so  direct  and  natural.  There  are  two  proposi- 
tions of  which  it  is  required  to  show  that  whenever 
the  first  is  true  the  second  is  true  ;  that  is,  the  first 
being  the  hypothesis  the  second  is  a  necessary  conclu- 
sion from  it,  whence  the  hypothesis  in  question,  and 
anything  contradictory  to,  or  inconsistent  with,  the 
conclusion  cannot  exist  together.  In  indirect  reason- 
ing, we  suppose  that,  the  original  hypothesis  existing 
and  being  true,  something  inconsistent  with  or  con- 
tradictory to  the  conclusion  is  true  also.  If  from  com- 
bining the  consequences  of  these  two  suppositions, 
something  evidently  erroneous  or  absurd  is  deduced, 
it  is  plain  that  there  is  something  wrong  in  the  as- 
sumptions. Now  care  is  taken  that  the  only  doubtful 
point  shall  be  the  one  just  alluded  to,  namely,  the 
supposition  that  one  proposition  and  the  contradictory 


GEOMETRICAL  REASONING.  227 

of  the  other  are  true  together.  This  then  is  incorrect, 
that  is,  the  first  proposition  cannot  exist  with  anything 
contradictory  to  the  second,  or  the  second  must  exist 
wherever  the  first  exists,  since  if  any  proposition  be 
not  true  its  contradictory  must  be  true,  and  vice  versa. 
This  is  rather  embarrassing  to  the  beginner,  who  finds 
that  he  is  required  to  admit,  for  argument's  sake,  a 
proposition  which  the  argument  itself  goes  to  destroy. 
But  the  difficulty  would  be  materially  lessened,  if  in- 
stead of  assuming  the  contradictory  of  the  second 
proposition  positively,  it  were  hypothetically  stated, 
and  the  consequences  of  it  asserted  with  the  verb 
"would  be,"  instead  of  "is."  For  example  :  suppose 
it  to  be  known  that  if  A  is  £,  then  C  must  be  D,  and 
it  is  required  to  show  indirectly  that  when  C  is  not  D, 
A  is  not  B.  This  put  into  the  form  in  which  such  a 
proposition  would  appear  in  most  elementary  works, 
is  as  follows. 

It  being  granted  that  if  A  is  B,  C  is  D,  it  is  re- 
quired to  show  that  when  C  is  not  D,  A  is  not  B.  If 
possible,  let  C  be  not  D,  and  let  A  be  B.  Then  by 
what  is  granted,  since  A  is  B,  C  is  D ;  but  by  hy- 
pothesis C  is  not  D,  therefore  both  C  is  D  and  is  not 
Dy  which  is  absurd  ;  that  is,  it  is  absurd  to  suppose 
that  C  is  not  D  and  A  is  B,  consequently  when  C  is 
not  D,  A  is  not  B.  The  following,  which  is  exactly 
the  same  thing,  is  plainer  in  its  language.  Let  C  be 
not  D.  Then  if  A  were  B,  C  would  be  D  by  the  prop- 
osition granted.  But  by  hypothesis  C  is  not  D,  etc. 


228  ON  THE  STUDY  OF  MATHEMATICS. 

This  sort  of  indirect  reasoning  frequently  goes  by  the 
name  of  reductio  ad  absurdum. 

In  all  that  has  gone  before  we  may  perceive  that 
the  validity  of  an  argument  depends  upon  two  distinct 
considerations,— (1)  the  truth  of  the  relations  assumed, 
or  represented  to  have  been  proved  before ;  (2)  the 
manner  in  which  these  facts  are  combined  so  as  to 
produce  new  relations  ;  in  which  last  the  reasoning 
properly  consists.  If  either  of  these  be  incorrect  in 
any  single  point,  the  result  is  certainly  false  ;  if  both 
be  incorrect,  or  if  one  or  both  be  incorrect  in  more 
points  than  one,  the  result,  though  not  at  all  to  be  de- 
pended on,  is  not  certainly  false,  since  it  may  happen 
and  has  happened,  that  of  two  false  reasonings  or 
facts,  or  the  two  combined,  one  has  reversed  the  effect 
of  the  other  and  the  whole  result  has  been  true ;  but 
this  could  only  have  been  ascertained  after  the  cor- 
rection of  the  erroneous  fact  or  reasoning.  The  same 
thing  holds  good  in  every  species  of  reasoning,  and  it 
must  be  observed,  that  however  different  geometrical 
argument  may  be  in  form  from  that  which  we  employ 
daily,  it  is  not  different  in  reality.  We  are  accus- 
tomed to  talk  of  mathematical  reasoning  as  above  all 
other,  in  point  of  accuracy  and  soundness.  This,  if 
by  the  term  reasoning  we  mean  the  comparing  together 
of  different  ideas  and  producing  other  ideas  from  the 
comparison,  is  not  correct,  for  in  this  view  mathemat- 
ical reasonings  and  all  other  reasonings  correspond 
exactly.  For  the  real  difference  between  mathematics 


GEOMETRICAL  REASONING.  2 29 

and  other  studies  in  this  respect  we  refer  the  student 
to  the  first  chapter  of  this  treatise. 

In  what  then,  may  it  be  asked,  does  the  real  ad- 
vantage of  mathematical  study  consist?  We  repeat 
again,  in  the  actual  certainty  which  we  possess  of  the 
truth  of  the  facts  on  which  the  whole  is  based,  and 
the  possibility  of  verifying  every  result  by  actual  meas- 
urement, and  not  in  any  superiority  which  the  method 
of  reasoning  possesses,  since  there  is  but  one  method 
of  reasoning.  To  pursue  the  illustration  with  which 
we  opened  this  work  (page  the  first),  suppose  this 
point  to  be  raised,  was  the  slaughter  of  Caesar  justifi- 
able or  not?  The  actors  in  that  deed  justified  them- 
selves by  saying,  that  a  tyrant  and  usurper,  who  med- 
itated the  destruction  of  his  country's  liberty,  made  it 
the  duty  of  every  citizen  to  put  him  to  death,  and  that 
Caesar  was  a  tyrant  and  usurper,  etc.  Their  reasoning 
was  perfectly  correct,  though  proceeding  on  premisses 
then  extensively,  and  now  universally,  denied.  The 
first  premiss,  though  correctly  used  in  this  reasoning, 
is  now  asserted  to  be  false,  on  the  ground  that  it  is 
the  duty  of  every  citizen  to  do  nothing  which  would, 
were  the  practice  universal,  militate  against  the  gen- 
eral happiness  ;  that  were  each  individual  to  act  upon 
his  own  judgment,  instead  of  leaving  offenders  to  the 
law,  the  result  would  be  anarchy  and  complete  de- 
struction of  civilisation,  etc.  Now  in  these  reasonings 
and  all  others,  with  the  exception  of  those  which  oc- 
cur in  mathematics,  it  must  be  observed  that  there 


230  ON  THE  STUDY  OF  MATHEMATICS. 

are  no  premisses  so  certain,  as  never  to  have  been 
denied,  no  first  principles  to  which  the  same  degree  of 
evidence  is  attached  as  to  the  following,  that  "no 
two  straight  lines  can  enclose  a  space."  In  mathe- 
matics, therefore,  we  reason  on  certainties,  on  notions 
to  which  the  name  of  innate  can  be  applied,  if  it  can 
be  applied  to  any  whatever.  Some,  on  observing  that 
we  dignify  such  simple  consequences  by  the  name  of 
reasoning,  may  be  loth  to  think  that  this  is  the  pro- 
cess to  which  they  used  to  attach  such  ideas  of  diffi- 
culty. There  may,  perhaps,  be  many  who  imagine 
that  reasoning  is  for  the  mathematician,  the  logician, 
etc.,  and  who,  like  the  Bourgeois  Gentilhomme,  may 
be  surprised  on  being  told,  that,  well  or  ill,  they  have 
been  reasoning  all  their  lives.  And  yet  such  is  the 
fact ;  the  commonest  actions  of  our  lives  are  directed 
by  processes  exactly  identical  with  those  which  enable 
us  to  pass  from  one  proposition  of  geometry  to  an- 
other. A  porter,  for  example,  who  being  directed  to 
carry  a  parcel  from  the  city  to  a  street  which  he  has 
never  heard  of,  and  who  on  inquiry,  finding  it  is  in 
the  Borough,  concludes  that  he  must  cross  the  water 
to  get  at  it,  has  performed  an  act  of  reasoning,  differ- 
ing nothing  in  kind  from  those  by  a  series  of  which, 
did  he  know  the  previous  propositions,  he  might  be 
convinced  that  the  square  of  the  hypothenuse  of  a 
right-angled  triangle  is  equal  to  the  sum  of  the  squares 
of  the  sides. 


CHAPTER  XV. 

ON  AXIOMS. 

EOMETRY,  then,  is  the  application  of  strict  logic 
to  those  properties  of  space  and  figure  which 
are  self-evident,  and  which  therefore  cannot  be  dis- 
puted. But  the  rigor  of  this  science  is  carried  one 
step  further ;  for  no  property,  however  evident  it  may 
be,  is  allowed  to  pass  without  demonstration,  if  that 
can  be  given.  The  question  is  therefore  to  demon- 
strate all  geometrical  truths  with  the  smallest  possible 
number  of  assumptions.  These  assumptions  are  called 
axioms,  and  for  an  axiom  it  is  requisite :  (1)  that  it 
should  be  self-evident ;  (2)  that  it  should  be  incapable 
of  being  proved  from  the  other  axioms.  In  fulfilling 
these  conditions,  the  number  of  axioms  which  are 
really  geometrical,  that  is,  which  have  not  equal  ref- 
erence to  Arithmetic,  is  reduced  to  two,  viz.,  two 
straight  lines  cannot  enclose  a  space,  and  through  a 
given  point  not  more  than  one  parallel  can  be  drawn 
to  a  given  straight  line.  The  first  of  these  has  never 
been  considered  as  open  to  any  objection;  it  has 


232  ON  THE  STUDY  OF  MATHEMATICS. 

always  passed  as  perfectly  self-evident.*  It  is  on  this 
account  made  the  proposition  on  which  are  grounded 
all  reasonings  relative  to  the  straight  line,  since  the 
definition  of  a  straight  line  is  too  vague  to  afford  any 
information.  But  the  second,  viz.,  that  through  a 
given  point  not  more  than  one  parallel  can  be  drawn 
to  a  given  straight  line,  has  always  been  considered 
as  an  assumption  not  self-evident  in  itself,  and  has 

*But  see  J.  B.  Stallo,  Concepts  and  Theories  of  Modern  Physics,  New  York, 
1884,  p.  242,  p.  208  et  seq.,  and  p.  248  et  seq.  For  popular  philosophical  dis- 
cussions of  the  subject  of  Axioms  generally,  in  the  light  of  modern  psychol- 
ogy and  pangeometry,  the  reader  may  consult  the  following  works :  Helm- 
holtz's  "Origin  and  Meaning  of  Geometrical  Axioms,"  Mind,  Vol.  III.,  p.  215, 
and  the  article  in  the  same  author's  Popular  Lectures  on  Scientific  Subjects, 
Second  Series,  London,  1881,  pp.  27-71 ;  W.  K.  Clifford's  Lectures  and  Essays, 
Vol.  I.,  p.  297,  p.  317;  Duhamel,  Des  Mlthodes  dans  les  Sciences  de  Raisonne- 
ment,  Part  2 ;  and  the  articles  "Axiom "  and  "Measurement "  in  the  Encyclo- 
pedia Britannica,  Vol.  XV.  See  also  Riemann's  Essay  on  the  Hypotheses 
Which  Lie  at  the  Basis  of  Geometry,  a  translation  of  which  is  published  in 
Clifford's  Works,  pp.  55-69.  For  part  of  the  enormous  technical  literature  of 
this  subject  cf.  Halsted's  Bibliography  of  Hyper-Space  and  Non-Euclidean 
Geometry,  American  Journal  of  Mathematics,  Vol.  I.,  pp.  261  et  seq.,  and  Vol. 
II.,  pp.  65  et  seq.  Much,  however,  has  been  written  subsequently  to  the  date 
of  the  last-mentioned  compilation,  and  translations  of  Lobachevski  and  Bo- 
lyai,  for  instance,  may  be  had  in  the  Neomonic  Series  of  Dr.  G.  B.  Halsted 
(Austin,  Texas).  A  full  history  of  the  theory  of  parallels  till  recent  times 
is  given  in  Paul  Stackel's  Theorie  der  Parallellinien  -von  Euklidbis  auf  Gauss 
(Leipsic,  1895).  Of  interest  are  the  essays  of  Prof.  J.  Delboeuf  on  The  Old 
and  the  New  Geometries  (Revue  Philosophique ,  1893-1895),  and  those  of  Profes- 
sor Poincare  and  of  other  controversialists  in  the  recent  volumes  of  the 
Revue  de  Mltaphysique  et  de  Morale,  where  valuable  bibliographical  refer- 
ences will  be  found  to  literature  not  mentioned  in  this  note.  See  also  P.  Tan- 
nery in  the  recent  volumes  of  the  Revue  gin  f rale  and  the  Revue philosophique ; 
Poincare  in  The  Monist  for  October,  1898,  and  B.  A.W.  Russell' s  Foundations 
of  Geometry  (Cambridge,  1897).  In  Grassmann' s  Ausdehnungslehre  (1844),  "  as- 
sumptions" and  "axioms"  are  replaced  by  purely  formal  (logical)  "predica- 
tions," which  presuppose  merely  the  consistency  of  mental  operations.  (See 
The  Open  Court,  Vol.  II.  p.  1464,  Grassmann,  "A  Flaw  in  the  Foundation  of 
Geometry,"  and  Hyde's  Directional  Calculus,  Ginn  &  Co.,  Boston).  Dr.  Paul 
Cams  in  his  Primer  of  Philosophy  (Chicago),  p.  51  et  seq.,  has  treated  the  sub- 
ject of  Axioms  at  length,  from  a  similar  point  of  view.  On  the  psychological 
side,  consult  Mach's  Analysis  of  the  Sensations  (Chicago,  1897),  and  the  biblio- 
graphical references  and  related  discussions  in  such  works  as  James's  Psy- 
chology and  Jodl's  Psychology  (Stuttgart,  1896).— Ed. 


AXIOMS. 


233 


therefore  been  called  the  defect  and  disgrace  of  geom- 
etry. We  proceed  to  place  it  on  what  we  conceive  to 
be  the  proper  footing. 

By  taking  for  granted  the  arithmetical  axioms  only, 
with  the  first  of  those  just  alluded  to,  the  following 
propositions  may  be  strictly  shown. 

I.  One  perpendicular,  and  only  one,  can  be  let  fall 
from  any  point  A  to  a.  given  line  CD.    Let  this  be  AB. 

II.  If  equal  distances  BC  and  BD  be  taken  on 
both  sides  of  B,  AC  and  AD  are  equal,  as  also  the 
angles  BAC  and  BAD. 


F 


Fig.  6. 


III.  Whatever  may  be  the  length  of  BC  and  BD, 
the  angles  BA  C  and  BAD  are  each  less  than  a  right 

igle. 

IV.  Through  A  a  line  may  be  drawn  parallel  to 
CD  (that  is,  by  definition,  never  meeting  CD,  though 
the  two  be  ever  so  far  produced),  by  drawing  any  line 
AD  and  making  the  angle  DAE  equal  to  the  angle 
ADB,  which  it  is  before  shown  how  to  do. 

From  proposition  IV.  we  should  at  first  see  no 


234  ON  THE  STUDY  OF  MATHEMATICS. 

reason  against  there  being  as  many  parallels  to  CD, 
to  be  drawn  through  A,  as  there  are  different  ways  of 
taking  AD,  since  the  direction  for  drawing  a  parallel 
to  CD  is,  "take  any  line  AD  cutting  CD  and  make 
the  angle  DAE  equal  to  ADB"  But  this  our  senses 
immediately  assure  us  is  impossible. 

It  appears  also  a  proposition  to  which  no  degree 
of  doubt  can  attach,  that  if  the  straight  line  AB,  pro- 
duced indefinitely  both  ways,  set  out  from  the  posi- 
tion AB  and  revolve  round  the  point  A,  moving  first 
towards  AE;  then  the  point  of  intersection  D  will 
first  be  on  one  side  of  B  and  afterwards  on  the  other, 
and  there  will  be  one  position  where  there  is  no  point 
of  intersection  either  on  one  side  or  the  other,  and  one 
such  position  only.  This  is  in  reality  the  assumption  of 
Euclid ;  for  having  proved  that  AE  and  BF  are  par- 
allel when  the  angles  BDA  and  DAE  are  equal,  or, 
which  is  the  same  thing,  when  EAD  and  ADF  are 
together  equal  to  two  right  angles,  he  further  assumes 
that  they  will  be  parallel  in  no  other  case,  that  is,  that 
they  will  meet  when  the  angles  EAD  and  ADF  are 
together  greater  or  less  than  two  right  angles ;  which 
is  really  only  assuming  that  the  parallel  which  he  has 
found  is  the  only  one  which  can  be  drawn.  The  re- 
maining part  of  his  axiom,  namely,  that  the  lines  AE 
and  DF,  if  they  meet  at  all,  will  meet  upon  that  side 
of  DA  on  which  the  angles  are  less  than  two  right 
angles,  is  not  an  assumption  but  a  consequence  of  his 
proposition  which  shows  that  any  two  angles  of  a 


AXIOMS.  235 

triangle  are  together  less  than  two  right  angles,  and 
which  is  established  before  any  mention  is  made  of 
parallels.  It  has  been  found  by  the  experience  of 
two  thousand  years  that  some  assumption  of  this  sort 
is  indispensable.  Every  species  of  effort  has  been 
made  to  avoid  or  elude  the  difficulty,  but  hitherto 
without  success,  as  some  assumption  has  always  been 
involved,  at  least  equal,  and  in  most  cases  superior, 
in  difficulty  to  the  one  already  made  by  Euclid.  For 
example,  it  has  been  proposed  to  define  parallel  lines 
as  those  which  are  equidistant  from  one  another  at 
every  point.  In  this  case,  before  the  name  parallel 
can  be  allowed  to  belong  to  any  thing,  it  must  be 
proved  that  there  are  lines  such  that  a  perpendicular 
to  one  is  always  perpendicular  to  the  other,  and  that 
the  parts  of  these  perpendiculars  intercepted  between 
the  two  are  always  equal.  A  proof  of  this  has  never 
been  given  without  the  previous  assumption  of  some- 
thing equivalent  to  the  axiom  of  Euclid.  Of  this  last, 
indeed,  a  proof  has  been  given,  but  involving  consid- 
erations not  usually  admitted  into  geometry,  though 
it  is  more  than  probable  that  had  the  same  come 
down  to  us,  sanctioned  by  the  name  of  Euclid,  it 
would  have  been  received  without  difficulty.  The 
Greek  geometer  confines  his  notion  of  equal  magni- 
tudes to  those  which  have  boundaries.  Suppose  this 
notion  of  equality  extended  to  all  such  spaces  as  can 
be  made  to  coincide  entirely  in  all  their  extent,  what- 
ever that  extent  may  be ;  for  example,  the  unbounded 


236 


ON  THE  STUDY  OF  MATHEMATICS. 


spaces  contained  between  two  equal  angles  whose 
sides  are  produced  without  end,  which  by  the  defini- 
tion of  equal  angles  might  be  made  to  coincide  entirely 
by  laying  the  sides  of  one  angle  upon  those  of  the 
other.  In  the  same  sense  we  may  say,  that,  one 
angle  being  double  another,  the  space  contained  by 
the  sides  of  the  first  is  double  that  contained  by  the 
sides  of  the  second,  and  so  on.  Now  suppose  two 


Fig.  7- 

lines  Oa  and  Ob,  making  any  angle  with  one  another, 
and  produced  ad  infinitum*  On  Oa  take  off  the  equal 
spaces  OP,  PQ,  QR,  etc.,  ad  infinitum,  and  draw  the 
lines  Pp,  Qq,  Rr,  etc.,  so  that  the  angles  OPp,  OQq, 
etc.,  shall  be  equal  to  one  another,  each  being  such 
as  with  bOP  will  make  two  right  angles.  Then  Ob, 
Pp,  Qq,  etc.,  are  parallel  to  one  another,  and  the  in- 

*  Every  line  in  this  figure  must  be  produced  ad  infinitum,  from  that  ex- 
tremity at  which  the  small  letter  is  placed. 


AXIOMS.  237 

finite  spaces  bOPp,  pPQq,  qQRr,  etc.,  can  be  made 
to  coincide,  and  are  equal.  Also  no  finite  number 
whatever  of  these  spaces  will  fill  up  the  infinite  space 
bOa,  since  OP,  PQ,  etc.,  may  be  contained  ad  infini- 
tum  upon  the  line  Oa.  Let  there  be  any  line  Ot,  such 
that  the  angles  tOP  and  pPO  are  together  less  than 
two  right  angles,  that  is,  less  than  bOP  and  pPO\ 
whence  tOP  is  less  than  bOP  and  tO  falls  between 
bO  and  a  O.  Take  the  angles  tOv,  vOwy  wOx,  each 
equal  to  bOt,  and  continue  this  until  the  last  line  Oz 
falls  beneath  Oa,  so  that  the  angle  bOz  is  greater  than 
bOa.  That  this  is  possible  needs  no  proof,  since  it  is 
manifest  that  any  angle  being  continually  added  to 
itself  the  sum  will  in  time  exceed  any  other  given  an- 
gle; again,  the  infinite  spaces  bOt,  tOv,  etc.,  are  all 
equal.  Now  on  comparing  the  spaces  bOt  and  bOPp, 
we  see  that  a  certain  number  of  the  first  is  more  than 
equal  to  the  space  bOa,  while  no  number  whatever  of 
the  second  is  so  great.  We  conclude,  therefore,  that 
the  space  bOt  is  greater  than  bOPp,  which  cannot  be 
unless  the  line  Ot  cuts  Pp  at  last ;  for  if  Ot  did  never 
cut  Pp,  the  space  bOt  would  evidently  be  less  than 
bOPp,  as  the  first  would  then  fall  entirely  within  the 
second.  Therefore  two  lines  which  make  with  a  third 
angles  together  less  than  two  right  angles  will  meet  if 
sufficiently  produced.  [See  Note  on  page  239.] 

This  demonstration  involves  the  consideration  of 
a  new  species  of  magnitude,  namely,  the  whole  space 
contained  by  the  sides  of  an  angle  produced  without 


238  ON  THE  STUDY  OF  MATHEMATICS. 

limit.  This  space  is  unbounded,  and  is  greater  than 
any  number  whatever  of  finite  spaces,  of  square  feet, 
for  example.  No  comparison,  therefore,  as  to  magni- 
tude can  be  instituted  between  it  and  any  finite  space 
whatever,  but  that  affords  no  reason  against  compar- 
ing this  magnitude  with  others  of  the  same  kind. 

Any  thing  may  become  the  subject  of  mathemati- 
cal reasoning,  which  can  be  increased  or  diminished 
by  other  things  of  the  same  kind ;  this  is,  in  fact,  the 
definition  given  of  the  term  magnitude;  and  geometri- 
cal reasoning,  in  all  other  cases  at  least,  can  be  ap- 
plied as  soon  as  a  criterion  of  equality  is  discovered. 
Thus  the  angle,  to  beginners,  is  a  perfectly  new  spe- 
cies of  magnitude,  and  one  of  whose  measure  they 
have  no  conception  whatever ;  they  see,  however,  that 
it  is  capable  of  increase  or  diminution,  and  also  that 
two  of  the  kind  can  be  equal,  and  how  to  discover 
whether  this  is  so  or  not,  and  nothing  more  is  neces- 
sary for  them.  All  that  can  be  said  of  the  introduc- 
tion of  the  angle  in  geometry  holds  with  some,  (to  us 
it  appears  an  equal  force,)  with  regard  to  these  unlim- 
ited spaces ;  the  two  are  very  closely  connected,  so 
much  so,  that  the  term  angle  might  even  be  defined 
as  "the  unlimited  space  contained  by  two  right  lines," 
without  alteration  in  the  truth  of  any  theorem  in  which 
the  word  angle  is  found.  But  this  is  a  point  which 
cannot  be  made  very  clear  to  the  beginner. 

The  real  difficulties  of  geometry  begin  with  the 
theory  of  proportion,  to  which  we  now  proceed.  The 


AXIOMS.  239 

points  of  discussion  which  we  have  hitherto  raised, 
are  not  such  as  to  embarrass  the  elementary  student, 
however  much  they  may  perplex  the  metaphysical  in- 
quirer into  first  principles.  The  theory  to  which  we 
are  coming  abounds  in  difficulties  of  both  classes. 


[NOTE  TO  PAGE  237. — The  demonstration  given  on  pp.  235- 
237  is  now  regarded  as  fallacious  by  mathematicians ;  the  consid- 
erations that  apply  to  finite  aggregates  not  being  transferable  to 
infinite  aggregates, — for  example,  it  is  not  true  for  infinite  aggre- 
gates that  the  part  is  always  less  than  the  whole  Even  Plato  is 
cited  for  the  assertion  that  equality  is  only  to  be  predicated  of 
finite  magnitudes.  See  the  modern  works  on  the  Theory  of  the 
Infinite.  The  demonstration  in  question  is  not  De  Morgan's,  but 
M.  Bertrand's.— Ed.} 


CHAPTER  XVI. 

ON  PROPORTION. 

TN  the  first  elements  of  geometry,  two  lines,  or  two 
•*•  surfaces,  are  mentioned  in  no  other  relation  to 
one  another  than  that  of  equality  or  non-equality. 
Nothing  but  the  simple  fact  is  announced  that  one 
magnitude  is  equal  to,  greater  than,  or  less  than  an- 
other, except  occasionally  when  the  sum  of  two  equal 
magnitudes  is  said  to  be  double  one  of  them.  Thus 
in  proving  that  two  sides  of  a  triangle  are  together 
greater  than  the  third,  the  fact  that  they  are  greater 
is  the  essence  of  the  proposition  ;  no  measure  is  given 
of  the  excess,  nor  does  anything  follow  from  the  theo- 
rem as  to  whether  it  is,  or  may  be,  small  or  great. 
We  now  come  to  the  doctrine  of  proportion  in  which 
geometrical  magnitude  is  considered  in  a  new  light. 
The  subject  has  some  difficulties,  which  have  been 
materially  augmented  by  the  almost  universal  use,  in 
this  country  at  least,*  of  the  theory  laid  down  in  the 
fifth  book  of  Euclid,  f  Considered  as  a  complete  con- 

*  In  England.  t  See  Todhunter's  Euclid  (Macmillan,  London).—  Ed. 


PROPORTION.  24! 

quest  over  a  great  and  acknowledged  difficulty  of  prin- 
ciple, this  book  of  Euclid  well  deserves  the  immortal- 
ity of  which  its  existence,  at  the  present  moment,  is 
the  guarantee  ;  nay,  had  the  speculations  of  the  math- 
ematician been  wholly  confined  to  geometrical  magni- 
tude, it  might  be  a  question  whether  any  other  notions 
would  be  necessary.  But  when  we  come  to  apply 
arithmetic  to  geometry,  it  is  necessary  to  examine  well 
the  primary  connexion  between  the  two ;  and  here 
difficulties  arise,  not  in  comprehending  that  connexion 
so  much  as  in  joining  the  two  sciences  by  a  chain  of 
demonstration  as  strong  as  that  by  which  the  propo- 
sitions of  geometry  are  bound  together,  and  as  little 
open  to  cavil  and  disputation. 

The  student  is  aware  that  before  pronouncing  upon 
the  connexion  of  two  lines  with  one  another,  it  is  ne- 
cessary to  measure  them,  that  is,  to  refer  them  to  some 
third  line,  and  to  observe  what  number  of  times  the 
third  is  contained  in  the  other  two.  Whether  the  two 
first  are  equal  or  not  is  readily  ascertained  by  the  use 
of  the  compasses,  on  principles  laid  down  with  the 
utmost  strictness  in  Euclid  and  other  elementary 
works.  But  this  step  is  not  sufficient ;  to  say  that  two 
lines  are  not  equal,  determines  nothing.  There  are 
an  infinite  number  of  ways  in  which  one  line  may  be 
greater  or  less  than  a  given  line,  though  there  is  only 
one  in  which  the  other  can  be  equal  to  the  given  one. 
We  proceed  to  show  how,  from  the  common  notion 


242 


ON  THE  STUDY  OF  MATHEMATICS. 


of  measuring  a  line,  the  more  strict  geometrical  method 
is  derived. 

To  measure  the  line  AB,  apply  to  it  another  line 
(the  edge  of  a  ruler),  which  is  divided  into  equal  parts 
(as  inches),  each  of  which  parts  is  again  subdivided 
into  ten  equal  parts,  as  in  the  figure.  This  division  is 
made  to  take  place  in  practice  until  the  last  subdivi- 
sion gives  a  part  so  small  that  anything  less  may  be 
neglected  as  inconsiderable.  Thus  a  carpenter's  rule 
is  divided  into  tenths  or  eighths  of  inches 
only,  while  in  the  tube  of  a  barometer  a 
process  must  be  employed  which  will 
mark  a  much  less  difference.  In  talking 
of  accurate  measurement,  therefore,  any- 
where but  in  geometry,  or  algebra,  we 
only  mean  accurate  as  far  as  the  senses 
are  concerned,  and  as  far  as  is  necessary 
for  the  object  in  view.  The  ruler  in  the 
figure  shows  that  the  line  AB  contains 
more  than  two  and  less  than  three  inches ;  and  closer 
inspection  shows  that  the  excess  above  two  inches  is 
more  than  sixth-tenths  of  an  inch,  and  less  than 
seven.  Here,  in  practice,  the  process  stops ;  for,  as 
the  subdivision  of  the  ruler  was  carried  only  to  tenths 
of  inches,  because  a  tenth  of  an  inch  is  a  quantity 
which  may  be  neglected  in  ordinary  cases,  we  may 
call  the  line  two  inches  and  six-tenths,  by  doing 
which  the  error  committed  is  less  than  one-tenth  of 
an  inch.  In  this  way  lines  may  be  compared  together 


Fig.  8. 


PROPORTION. 

with  a  common  degree  of  correctness ;  but  this  is  not 
enough  for  the  geometer.  His  notions  of  accuracy 
are  not  confined  to  tenths  or  hundredths,  or  hundred- 
millionth  parts  of  any  line,  however  small  it  may  be 
at  first.  The  reason  is  obvious ;  for  although  to  suit 
the  eye  of  the  generality  of  readers,  figures  are  drawn 
in  which  the  least  line  is  usually  more  than  an  inch, 
yet  his  theorems  are  asserted  to  remain  true,  even 
though  the  dimensions  of  the  figure  are  so  far  dimin- 
ished as  to  make  the  whole  imperceptible  in  the 
strongest  microscope.  Many  theorems  are  obvious 
upon  looking  at  a  moderately-sized  figure ;  but  the 
reasoning  must  be  such  as  to  convince  the  mind  of 
their  truth  when,  from  excessive  increase  or  diminu- 
tion of  the  scale,  the  figures  themselves  have  past  the 
boundary  even  of  imagination.  The  next  step  in  the 
process  of  measurement  is  as  follows,  and  will  lead  us 
to  the  great  and  peculiar  difficulty  of  the  subject. 

The  inch,  the  foot,  and  the  other  lengths  by  which 
we  compare  lines  with  one  another,  are  perfectly  arbi- 
trary. There  is  no  reason  for  their  being  what  they 
are,  unless  we  adopt  the  commonly  received  notion 
that  our  inch  is  derived  from  our  Saxon  ancestors, 
who  observed  that  a  barley-corn  is  always  of  the  same 
length,  or  nearly  so,  and  placed  three  of  them  together 
as  a  common  standard  of  measure,  which  they  called 
an  inch.  Any  line  whatever  may  be  chosen  as  the 
standard  of  measure,  and  it  is  evident  that  when  two 
or  more  lines  are  under  consideration,  exact  compari- 


244  ON  THE  STUDY  OF  MATHEMATICS. 

sons  of  their  lengths  can  only  be  obtained  from  a  line 
which  is  contained  an  exact  number  of  times  in  them 
all.  For  even  exact  fractional  measures  are  reduced 
to  the  same  denominator,  in  order  to  compare  their 
magnitudes.  Thus,  two  lines  which  contain  T2T  and  ^ 
of  a  foot,  are  better  compared  by  observing  that  ^ 
and  f  being  ^  and  %%,  the  given  lines  contain  one 
77th  part  of  a  foot  14  and  33  times  respectively.  Any 
line  which  is  contained  an  exact  number  of  times  in 
another  is  called  in  geometry  a  measure  of  it,  and  a 
common  measure  of  two  or  more  lines  is  that  which 
is  contained  an  exact  number  of  times  in  each. 

Again,  a  line  which  is  measured  by  another  is  called 
a  multiple  of  it,  as  in  arithmetic. 

The  same  definition,  mutatis  mutandis,  applies  to 
surfaces,  solids,  and  all  other  magnitudes  ;  and  though 
in  our  succeeding  remarks  we  use  lines  as  an  illustra- 
tion, it  must  be  recollected  that  the  reasoning  applies 
equally  to  every  magnitude  which  can  be  made  the 
subject  of  calculation. 

In  order  that  two  quantities  may  admit  of  com- 
parison as  to  magnitude,  they  must  be  of  the  same 
sort ;  if  one  is  a  line,  the  other  must  be  a  line  also. 
Suppose  two  lines  A  and  B  each  of  which  is  measured 
by  the  line  C ';  the  first  containing  it  five  times  and 
the  second  six.  These  lines  A  and  B,  which  contain 
the  same  line  C  five  and  six  times  respectively,  are 
said  to  have  to  one  another  the  ratio  of  five  to  six,  or 
to  be  in  the  proportion  of  five  to  six.  If  then  we  de- 


PROPORTION.  245 

note  the  first  by  A,*  and  the  second  by  B,  and  the 
common  measure  by  C,  we  have 

A  =  5C,     or     QA=3QC, 
B  =  6C,     or     5^  =  30C, 
whence  §A  =  §B,     or     6A—  5£  =  Q. 
Generally,  when  mA  —  nJ?  =  Q,  the  lines,  or  what- 
ever they  are,  represented  by  A  and  B,  are  said  to  be 
in  the  proportion  of  n  to  m,  or  to  have  the  ratio  of  n 
to  m. 

Let  there  be  two  other  magnitudes  P  and  Q,  of 
the  same  kind  with  one  another,  either  differing  from 
the  first  in  kind  or  not,  (thus  A  and  B  may  be  lines, 
and  P  and  Q  surfaces,  etc.,)  and  let  them  contain  a 
common  measure  R,  just  as  A  and  B  contain  C,  viz. : 
Let  P  contain  R  five  times,  and  let  Q  contain  R  six 
times,  we  have  by  the  same  reasoning 
6^—5(2  =  0, 

and  P  and  Q,  being  also  in  the  ratio  of  five  to  six,  as 
well  as  A  and  B,  are  said  to  be  proportional  to  A  and 
B,  which  is  denoted  thus 

A'.B'.'.P:Q, 
by  which  at  present  all  we  mean  is  this,  that  there  are 

*  The  student  must  distinctly  understand  that  the  common  meaning  of 
algebraical  terms  is  departed  from  in  this  chapter,  wherever  the  letters  are 
large  instead  of  small.  For  example,  A,  instead  of  meaning  the  number  of 
units  of  some  sort  or  other  contained  in  the  line  A,  stands  for  the  line  A  itself, 
and  mA  (the  small  letters  throughout  meaning  whole  numbers}  stands  for  the 
line  made  by  taking  A,  m  times.  Thus  such  expressions  as  mA  +  B,  mA  —  nB, 

etc.,  are  the  only  ones  admissible.    AB,  — ,  A2,  etc.,  are  unmeaning,  while  — 

£>  m 

is  the  line  which  is  contained  m  times  in  A,  or  the  ;«th  part  of  A.  The  capital 
letters  throughout  stand  for  concrete  quantities,  not  for  their  representations 
in  abstract  numbers, 


246 


ON  THE  STUDY  OF  MATHEMATICS. 


some  two  whole  numbers  m  and  n  such  that,  at  the 

same  time 

mA  —  n£  =  Q, 


Nothing  more  than  this  would  be  necessary  for  the 
formation  of  a  complete  theory  of  proportion,  if  the 
common  measure,  which  we  have  supposed  to  exist 
in  the  definition,  did  always  really  exist.  We  have, 

however,  no  right  to  as- 
sume that  two  lines  A 
and  B,  whatever  may  be 
their  lengths,  both  con- 
tain some  other  line  an 
exact  number  of  times. 
We  can,  moreover,  pro- 
duce a  direct  instance  in 
which  two  lines  have  no 
common  measure  what- 
ever, in  the  following 
manner. 


Fig  9. 


Let  ABC  be  an  isosceles  right-angled  triangle,  the 
side  BC  and  the  hypothenuse  have  no  common  meas- 
ure whatever.  If  possible  let  D  be  a  common  meas- 
ure of  BC  and  AB ;  let  BC  contain  D,  n  times,  and 
let  AB  contain  D,  m  times.  Let  E  be  the  square  de 
scribed  on  D.  Then  since  AB  contains  D,  m  times, 
the  square  described  on  AB  contains  E,  m  X  m  or  m* 
times.  Similarly  the  square  described  on  BC  contains 
E  n  X  n  or  n2  times.  But,  because  AB  is  an  isosce- 


PROPORTION.  247 

les  right-angled  triangle,  the  square  on  AB  is  double 
that  on  BC,  whence  m  X  m  =  2  (n  X  »)  or  m2  =  2n2.  To 
prove  the  impossibility  of  this  equation  (when  m  and 
n  are  whole  numbers),  observe  that  m2  must  be  an 
even  number,  since  it  is  twice  the  number  n2.  But 
my^m  cannot  be  an  even  number  unless  m  is  an  even 
number,  since  an  odd  number  multiplied  by  itself 
produces  an  odd  number.*  Let  m  (which  has  been 
shown  to  be  even)  be  double  m'  or  m  =  2m'.  Then 
2m'  X  2m'  =  2n2  or  4m'2  =  2n2  or  n2  =  2m'2.  By  repeat- 
ing the  same  reasoning  we  show  that  n  is  even.  Let 
it  be  2n'.  Then  2n'  X  2»'  =  2m'2  or  m'2  =  2n'2.  By  the 
same  reasoning  m'  and  n'  are  both  even,  and  so  on  ad 
infinitum.  This  reasoning  shows  that  the  whole  num- 
bers which  satisfy  the  equation  n2  =  2m2  (if  such  there 
be)  are  divisible  by  2  without  remainder,  ad  infinitum. 
The  absurdity  of  such  a  supposition  is  manifest:  there 
are  then  no  such  whole  numbers,  and  consequently  no 
common  measure  to  BA  and  BC. 

Before  proceeding  any  further,  it  will  be  necessary 
to  establish  the  following  proposition. 

If  the  greater  of  two  lines  A  and  B  be  divided  into 
m  equal  parts,  and  one  of  these  parts  be  taken  away ; 
if  the  remainder  be  then  divided  into  m  equal  parts, 
and  one  of  them  be  taken  away,  and  so  on, — the  re- 


*  Every  odd  number,  when  divided  by  2,  gives  a  remainder  r,  and  is  there- 
fore of  the  form  2/  +  i  where/  is  a  whole  number.  Multiply  <zp  +  i  by  itself, 
which  gives  4/2  +  4/  +  i,  or  2  (zp2 +  zp}-\-i,  which  is  an  odd  number,  since, 
when  divided  by  2,  it  gives  the  quotient  2/2  -(-  2/,  a  whole  number,  and  the 
remainder  i. 


248  ON  THE  STUDY  OF  MATHEMATICS. 

mainder  of  the  line  A  shall  in  time  become  less  than 
the  line  B,  how  small  soever  the  line  B  may  be. 

Take  a  line  which  is  less  than  B,  and  call  it  C.  It 
is  evident  that,  by  a  continual  addition  of  the  same 
quantity  to  C,  this  last  will  come  in  time  to  exceed  A\ 
and  still  moire  will  it  do  so  if  the  quantity  added  to  C 
be  increased  at  each  step.  To  simplify  the  proof  we 
suppose  that  20  is  the  number  of  equal  parts  into 
which  A  and  its  remainders  are  successively  divided, 
so  that  19  out  of  the  20  parts  remain  after  subtraction. 

Divide  C  into  19  equal  parts  and  add  to  C  a.  line 
equal  to  one  of  these  parts.  Let  the  length  of  C,  so 
increased,  be  C'.  Divide  C'  into  19  equal  parts  and 
let  C',  increased  by  its  19th  part,  be  C".  Now,  since 
we  add  more  and  more  each  time  to  C,  in  forming  C', 
C",  etc,  we  shall  in  time  exceed  A.  Let  this  have 
been  done,  and  let  D  be  the  line  so  obtained,  which 
is  greater  than  A.  Observe  now  that  C'  contains  19, 
and  C",  20  of  the  same  parts,  whence  C'  is  made  by 
dividing  C"  into  20  parts  and  removing  one  of  them. 
The  same  of  all  the  rest.  Therefore  we  may  return 
from  D  to  C  by  dividing  D  into  20  parts,  removing 
one  of  them,  and  repeating  the  process  continually. 
But  C  is  less  than  B  by  hypothesis.  If  then  we  can, 
by  this  process,  reduce  D  below  B,  still  more  can  we 
do  so  with  A,  which  is  less  than  D,  by  the  same 
method. 

This  depends  on  the  obvious  truth,  that  if,  at  the 
end  of  any  number  of  subtractions  (Z>  being  taken), 


PROPORTION.  249 

we  have  left  —  Z>,  at  the  end  of  the  same  number  of 

9  P 

subtractions  (A  being  taken),  we  shall  have  —A,  since 

the  method  pursued  in  both  cases  is  the  same.     But 

since  A  is  less  than  D,  —A  is  less  than  -£>,  which  be- 

?       p  V 

comes  equal  to  C,  therefore  -  A  becomes  less  than  C.* 

We  now  resume  the  isosceles  right-angled  triangle. 
The  lines  BC  and  AB,  which  were  there  shown  to 
have  no  common  measure,  are  called  incommensurable 
quantities,  and  to  their  existence  the  theory  of  pro- 
portion owes  its  difficulties.  We  can  nevertheless 
show  that  A  and  B  being  incommensurable,  a  line  can 
be  found  as  near  to  B  as  we  please,  either  greater  or 
less,  which  is  commensurable  with  A.  Let  D  be  any 
line  taken  at  pleasure,  and  therefore  as  small  as  we 
please.  Divide  A  into  two  equal  parts,  each  of  those 
parts  into  two  equal  parts,  and  so  on.  We  shall  thus 
at  last  find  a  part  of  A  which  is  less  than  D.  Let  this 
part  be  E,  and  let  it  be  contained  m  times  in  A.  In 
the  series  E,  2E,  3E,  etc.,  we  shall  arrive  at  last  at 
two  consecutive  terms,  pE  and  (/+  1)^  of  which  the 
first  is  less,  and  the  second  greater  than  B.  Neither  of 
these  differs  from  B  by  so  much  as  E ;  still  less  by  so 
much  as  D ;  and  both  pE  and  (/-f- 1)2?  are  commen- 

*  Algebraically,  let  a  be  the  given  line,  and  let  — th  part  of  the  remainder 

191  a 

be  removed  at  every  subtraction.    The  first  quantity  taken  away  is  —  and  the 
ft  (         i  \  ^* 

remainder  a or  all ),  whence  the  second  quantity  removed  is 

at          \\       m  ^        m/       t  a\  f          i  \  /  i  \  2 

—  1  i )  ,   and  the  remainder  [a  —  —  II  (i I  or  a   I  i I    . 

m  ^          m'  V  •»/ A          m)  \          m> 

Similarly,  the  nth  remainder  is  a  (i -)    .    Now,  since  i —  is  less 

^         in  '  m 

than  unity,  its  powers  decrease,  and  a  power  of  so  great  an  index  may  be 
taken  as  to  be  less  than  any  given  quantity. 


250  ON  THE  STUDY  OF  MATHEMATICS. 

surable  with  A,  that  is  with  mE,  since  E  is  a  common 
measure  of  both.  If  therefore  A  and  B  are  incommen- 
surable, a  third  magnitude  can  be  found,  either  greater 
or  less  than  B,  differing  from  B  by  less  than  a  given 
quantity,  which  magnitude  shall  be  commensurable 
with  A. 

We  have  seen  that  when  A  and  B  are  incommen- 
surable, there  are  no  whole  values  of  m  and  n,  which 
will  satisfy  the  equation  #2-^  —  nB  =  Q',  nevertheless, 
we  can  prove  that  values  of  m  and  n  can  be  found 
which  will  make  mA  —  nB  less  than  any  given  magni- 
tude C,  of  the  same  kind,  how  small  soever  it  may  be. 
Suppose,  that  for  certain  values  of  m  and  n,*  we  find 
mA  —  nB  =  E,  and  let  the  first  multiple  of  E,  which 
is  greater  than  B,  be^E,  so  that  pE  =  B  +  E'  where 
E  is  less  than  E,  for  were  it  greater,  (/  —  !)-£>  or 
pE  —  E,  which  is  B  +  (E'  —  E),  would  be  greater 
than  B,  which  is  against  the  supposition. 

The  equation  mA  —  nB  =  E  gives 

/  mA  —  p  nB  —  pE  =  B  +  E', 
whence 


*It  is  necessary  here  to  observe,  that  in  speaking  of  the  expression  mA  — 
nB  we  more  frequently  refer  to  its  form  than  to  any  actual  value  of  it,  derived 
from  supposing  m  and  n  to  have  certain  known  values.  When  we  say  that 
mA  —  .nB  can  be  made  smaller  than  C,  we  mean  that  some  values  can  be 
given  to  m  and  n  such  that  mA  —  «5<  C,  or  that  some  multiple  of  B  subtracted 
from  some  multiple  of  A  is  less  than  C.  The  following  expressions  are  all  of 
the  same  form,  viz.,  that  of  some  multiple  of  B  subtracted  from  some  mul- 
tiple of  A  : 

mA  —  nB 

mpA—lnp  +  i)  B 
zmA  —  ^mB,  etc.,  etc. 


PROPORTION.  251 

Let 

pm=m'  and  p  n  -\-  1  =  «', 

whence 

m'A  —  riB  =  E'. 

We  have  therefore  found  a  difference  of  multiples 
which  is  less  than  E.  Let  'p'E'  be  the  first  multiple 
of  E'  which  is  greater  than  B,  where  p'  must  be  at 
least  as  great  as  /,  since  E  being  greater  than  E',  it 
cannot  take  more*  of  E  than  of  E'  to  exceed  B.  Let 


then,  as  before, 

m'p'A  —  (rip'  +  1)  B  =  .£",  • 

or 

w"^  —  n"  £  =  £"-, 

we  have  therefore  still  further  diminished  the  differ- 
ence of  the  multiples  ;  and  the  process  may  be  re- 
peated any  number  of  times  ;  it  only  remains  to  show 
that  the  diminution  may  proceed  to  any  extent. 

This  will  appear  superfluous  to  the  beginner,  who 
will  probably  imagine  that  a  quantity  diminished  at 
every  step,  must,  by  continuing  the  number  of  steps, 
at  last  become  as  small  as  we  please.  Nevertheless 
if  any  number,  as  10,  be  taken  and  its  square  root  ex- 
tracted, and  the  square  root  of  that  square  root,  and 
so  on,  the  result  will  not  be  so  small  as  unity,  although 
ten  million  of  square  roots  should  have  been  extracted. 
Here  is  a  case  of  continual  diminution,  in  which  the 
diminution  is  not  without  limit.  Again,  from  the  point 

*It  may  require  as  many.    Thus  it  requires  as  many  of  7  as  of  8  to  exceed 
33,  though  7  is  less  than  8. 


252  ON  THE  STUDY  OF  MATHEMATICS. 

D  in  the  line  AB  draw  DE,  making  an  angle  with 
AB  less  than  half  a  right  angle.  Draw  BE  perpen- 
dicular to  AB,  and  take  BC=BE.  Draw  CF  perpen- 
dicular to  AB,  and  take  CC'=CF,  and  so  on.  The 
points  C,  C',  C",  etc.,  will  always  be  further  from  A 
than  D  is;  and  all  the  lines  AC,  AC',  AC",  etc., 
though  diminished  at  every  step,  will  always  remain 
greater  than  AD.  Some  such  species  of  diminution, 
for  anything  yet  proved  to  the  contrary,  may  take 

place  in  mA  —  nB. 

E 


D     C"  C'      C 

Fig.  10. 


To  compare  the  quantities  E,  E',  etc.,  we  have 

the  equations 

pE  =  B  +  E' 

/£'  =  B  +  E" 


etc.  etc. 

The  numbers/,  /,  /',  etc.,  do  not  diminish;  the 
lines  E,  E',  E",  etc.,  diminish  at  every  step.  If  then 
we  can  show  that/,  p'  ,  etc.,  can  only  remain  the  same 
for  a  finite  number  of  steps,  and  must  then  increase, 
and  after  the  increase  can  only  remain  the  same  for 
another  finite  number  of  steps,  and  then  must  increase 
again,  and  so  on,  we  show  that  the  process  can  be 
continued,  until  one  of  them  is  as  great  as  we  please  ; 


PROPORTION.  253 

let  this  be/(*},  where  z  is  not  an  exponent,  but  marks 
the  number  which  our  notation  will  have  reached,  and 
indicates  the  (z  -f  l)th  step  of  the  process.  Let  E(z)  be 
the  corresponding  remainder  from  the  former  step. 
Then,  since  p™E^  is  the  first  multiple  of  E(z\  which 
exceeds  the  given  quantity  B,  if  pw  can  be  as  great  as 
we  please,  E(z)  can  be  as  small  as  we  please.  To  show 
that/^  can  be  as  great  as  we  please,  observe,  that/, 
/',  /",  etc.,  must  remain  the  same,  or  increase,  since, 
as  appears  from  their  method  of  formation,  they  can- 
not diminish.  Let  them  remain  the  same  for  some 
steps,  that  is,  let  p  =/'  =/',  etc.  The  equations  be- 
come 


etc.         etc. 
Then  by  subtraction, 

E'  —  E"  =p(E  —  E'} 

E"  —E'"  =p(E'  —  E'")=pp(E—E') 

E"'—E""=p  (E"  —  E'"}  =ppp  (E  —  E"} 

etc.  etc. 

Now, 

E-E"  =E-E'+E'-E"  =(^-^')(l+/) 

E-E"r=E-E'+E'-E"+E"-E'"=(E-E'}(\+p+p^ 

etc.  etc.  etc. 

Generally, 
E  _  E(^  =  E  _  E'  +  E'  —  E"  -\- 


254  ON  THE  STUDY  OF  MATHEMATICS. 

which  is  derived  from  w  steps  of  the  process.  Now, 
if  this  can  go  on  ad  infinitum,  it  can  go  on  until  1  + 

/+/2  + +/7""1  is   as   great   as  we   please;  for, 

since  p  is  not  less  than  unity,  the  continual  addition 
of  its  powers  will,  in  time,  give  a  sum  exceeding  any 
given  number.  This  is  absurd,  from  the  step  at  which 
1  -\-p  +/2  -f- . . .  -\-p'w~l  becomes  greater  than  the  num- 
ber of  times  which  E — E'  is  contained  in  E ;  for, 
from  the  above  equation,  E  —  E1  is  contained  in 
E—E^\  1  +  /+/2  _j_  _  e  _|_ pv- 1  times  .  and  it  ;s  con. 

tradictory  to  suppose  that  E — E'  should  be  contained 
in  E — E^  more  times  than  it  is  contained  in  E. 

To  take  an  example  :  suppose  that  B  is  55  feet, 
and  E  is  54  feet ;  the  first  equation  is 
2  X  54'=  55'+  53', 

where  E' =  53'  and  E — .£"  =  !',  and  is  contained  in 
E  54  times.  If,  then,  we  continue  the  process,  2  can- 
not maintain  its  present  place  through  so  many  steps 
of  the  process  as  will,  if  the  same  number  of  terms  be 
taken,  give  l  +  2  +  22  +  23+,  etc.,  greater  than  54; 
that  is,  it  cannot  be  the  same  for  six  steps.  And  we 
find,  on  actually  performing  the  operations, 

2x54^55'+  53' 

2x53'=  55'+ 51' 

2x51'=  55'+ 47' 

2x47'=  55'+ 39' 

2x39'=  55'+ 23' 

3x23'=  55'+ 14' 
We  do  not  say  that  /,  /',   etc.,   will  remain  the 


PROPORTION.  255 

same  until  1+/H-/2  -j-  .  .  .  would  be  greater  than  the 
number  of  times  which  E  contains  E  —  E',  but  only 
that  they  cannot  remain  the  same  longer.  By  repeti- 
tion of  the  same  process,  we  can  show  that  a  further 
and  further  increase  must  take  place,  and  so  on  until 
we  have  attained  a  quantity  greater  than  any  given 
one.  And  it  has  already  been  shown  to  be  a  conse- 
quence of  this,  that  mA  —  nB  can  be  diminished  to 
any  extent  we  please.  Similarly  it  may  be  shown  that 
when  A  and  B  are  incommensurable,  mA  —  nB  may 
be  brought  as  near  as  we  please  to  any  other  quantity 
C,  of  the  same  kind  as  A  and  B,  so  as  not  to  differ 
from  Cby  so  much  as  a  given  quantity  E.  For  let  m 
and  n  be  taken,  by  the  last  case,  so  that  mA  —  nB  may 
be  less  than  E,  and  let  mA  —  nB,  in  this  case,  be 
equal  to  E'.  Let  C  lie  between  pE'  and  (/  -f-  !)£', 
neither  of  which  can  differ  from  C  by  so  much  as  E't 
and  therefore  not  by  so  much  as  E.  Then  since 


therefore  pmA  —  pnB=pE', 

and         (p+l)mA  —  (p  +  \}n£  =  (p 

Both  which  last  expressions  differ  from  C  by  a  quan- 

tity less  than  E,  the  first  being  less  and  the  second 

greater  than  C,  and  both  are  of  the  form  mA  —  nB,  m 

and  n  being  changed  for  other  numbers. 

The  common  ideas  of  proportion  are  grounded 
entirely  upon  the  false  notion  that  all  quantities  of 
the  same  sort  are  commensurable.  That  the  supposi- 
tion is  practically  correct,  if  there  are  any  limits  to 


256  ON  THE  STUDY  OF  MATHEMATICS. 

the  senses,  may  be  shown,  for  let  any  quantity  be  re- 
jected as  imperceptible,  then  since  a  quantity  can  be 
found  as  near  to  B  as  we  please,  which  is  commensur- 
able with  A,  the  difference  between  B  and  its  approx- 
imate commensurable  magnitude,  may  be  reduced  be- 
low the  limits  of  perceptible  quantity.  Nevertheless, 
inaccuracy  to  some  extent  must  infest  all  general  con- 
clusions drawn  from  the  supposition  that  A  and  B 
being  two  magnitudes,  whole  numbers,  m  and  «,  can 
always  be  found  such  that  mA  —  nB  =  0.  We  have 
shown  that  this  can  be  brought  as  near  to  the  truth  as 
we  please,  since  mA  —  nB  can  be  made  as  small  as  we 
please.  This,  however,  is  not  a  perfect  answer,  at 
least  it  wants  the  unanswerable  force  of  all  the  pre- 
ceding reasonings  in  geometry.  A  definition  of  pro- 
portion should  therefore  be  substituted,  which,  while 
it  reduces  itself,  in  the  case  of  commensurable  quan- 
tities to  the  one  already  given,  is  equally  applicable 
to  the  case  of  incommensurables.  We  proceed  to  ex- 
amine the  definition  already  given  with  a  view  to  this 
object. 

Resume  the  equations 

—  nB  =  Q,  or  A=-B 
m 

—nQ  =  b,  or  P=  -  Q 

m 

If  we  take  any  other  expression  of  the  same  sort 
— ; ,B  and  — ,Q,  it  is  plain  that,  according  as  the  arith- 
metical fraction  —  is  greater  than,  equal  to,  or  less 


PROPORTION. 


257 


than  — ;  so  will  — B  be  greater  than,  equal  to.  or  less 
m  m 

than  — -,B,   and  the  same  of—  Q  and — ,Q.     Let  the 
m  mm 

symbol 


be  the  abbreviation  of  the  following  sentence :  "when 
x  is  greater  than  y,  z  is  greater  than  w ;  when  x  is 
equal  to  y,  z  is  equal  to  w ;  when  x  is  less  than  y,  z  is 
less  than  «/. "  The  following  conclusions  will  be  evi- 
dent: 
If 

* i >  :=<(^an  ')>=<1/ 

Then 

'l^       ^\d 

M>       <1/  W 

And  from  the  first  of  these  alone  it  follows  that 


ma 
nc 


tnb 
nd 


(2) 


We  have  just  noticed  the  following  : 


n 

n'                 n 

n' 

m 

>  =  < 

^      and    n 

>  =  < 

~^' 

m     . 

~riB            ~m@ 

.  m' 

Therefore  (1) 

n 

t 

n' 

m 
n 

>>  =  < 

•«*    or    Ap}>  =  <- 

If 

Therefore  (2) 


m  A 
m'P 


n'Q 


258  ON  THE  STUDY  OF  MATHEMATICS. 

Or,  if  four  magnitudes  are  proportional,  according  to 
the  common  notion,  it  follows  that  the  same  multiples 
of  the  first  and  third  being  taken,  and  also  of  the  sec- 
ond and  fourth,  the  multiple  of  the  first  is  greater 
than,  equal  to,  or  less  than,  that  of  the  second,  ac- 
cording as  that  of  the  third  is  greater  than,  equal  to, 
or  less  than,  that  of  the  fourth.  This  property*  ne- 
cessarily follows  from  the  equations 
mA  —  n£  =  Q 


but  it  does  not  therefore  follow  that  the  equations  are 
necessary  consequences  of  the  property,  since  the  lat- 
ter may  possibly  be  true  of  incommensurable  quanti- 
ties, of  which,  by  definition,  the  former  is  not.  The 
existence  of  this  property  is  Euclid's  definition  of  pro- 
portion :  he  says,  let  four  magnitudes,  two  and  two, 
of  the  same  kind,  be  called  proportional,  when,  if  equi- 
multiples be  taken  of  the  first  and  third,  etc.,  repeat- 
ing the  property  just  enunciated.  What  is  lost  and 
gained  by  adopting  Euclid's  definition  may  be  very 
simply  stated  ;  the  gain  is  an  entire  freedom  from  all 
the  difficulties  of  incommensurable  quantities,  and 
even  from  the  necessity  of  inquiring  into  the  fact  of 
their  existence,  and  the  removal  of  the  inaccuracy  at- 
tending the  supposition  that,  of  two  quantities  of  the 
same  kind,  each  is  a  determinate  arithmetical  fraction 
of  the  other  ;  on  the  other  hand,  there  is  no  obvious 

*It  would  be  expressed  algebraically  by  saying  that  if  mA  —  nB  and 
mP  —  nQ  are  nothing  for  the  same  values  of  m  and  «,  they  are  either  both 
positive  or  both  negative,  for  every  other  value  of  m  and  n. 


PROPORTION.  259 

connexion  between  Euclid's  definition  and  the  ordi- 
nary and  well-established  ideas  of  proportion ;  the 
definition  itself  is  made  to  involve  the  idea  of  infinity, 
since  all  possible  multiples  of  the  four  quantities  enter 
into  it;  and  lastly,  the  very  existence  of  the  four 
quantities,  called  proportional,  is  matter  for  subse- 
quent demonstration,  since  to  a  beginner  it  cannot 
but  appear  very  unlikely  that  there  are  any  magni- 
tudes which  satisfy  the  definition.  The  last  objection 
is  not  very  strong,  since  the  learner  could  read  the 
first  proposition  of  the  sixth  book  immediately  after 
the  definition,  and  would  thereby  be  convinced  of  the 
existence  of  proportionals ;  the  rest  may  be  removed 
by  showing  another  definition,  -more  in  consonance 
with  common  ideas,  and  demonstrating  that,  if  four 
magnitudes  fall  under  either  of  these  definitions,  they 
fall  under  the  other  also.  The  definition  which  we 
propose  is  as  follows:  "Four  magnitudes,  A,  B,  P, 
and  Q,  of  which  B  is  of  the  same  kind  as  A,  and  Q 
as  P,  are  said  to  be  proportional,  if  magnitudes  B-\-  C 
and  Q-\-  R  can  be  found  as  near  as  we  please  to  B  and 
Q,  so  that  A,  B  +  C,  P  and  Q  +  R,  are  proportional 
according  to  the  common  notion,  that  is,  if  whole 
numbers  m  and  n  can  satisfy  the  equations 

mA  —  n(B-\-  C)=0 

mP—n(  Q  +  R)=§. 

We  have  now  to  show  that  Euclid's  definition  fol- 
lows from  the  one  just  given,  and  also  that  the  last 
follows  from  Euclid's,  that  is,  if  there  are  four  magni- 


260  ON  THE  STUDY  OF  MATHEMATICS. 

tudes  which  fall  under  either  definition,  they  fall  un- 
der the  other  also.     Let  us  first  suppose  that  Euclid's 
definition  is  true  of  A,  B,  P,  and  Q,  so  that 
mA 


This  being  true,  it  will  follow  that  we  can  take  m  and 
n,  so  as  not  only  to  make  mA  —  nB  less  than  a  given 
magnitude  E,  which  may  be  as  small  as  we  please, 
but  also  so  that  mP  —  nQ  shall  at  the  same  time  be 
less  than  a  given  magnitude  F,  however  small  this 
last  may  be.  For  if  not,  while  m  and  n  are  so  taken 
as  to  make  mA  —  nB  less  than  E  (which  it  has  been 
proved  can  be  done,  however  small  E  may  be)  sup- 
pose, if  possible,  that  the  same  values  of  m  and  n  will 
never  make  mP  —  nQ  less  than  some  certain  quantity 
F,  and  let  pF  be  the  first  multiple  of  F  which  exceeds 
Q,  and  also  let  E  be  taken  so  small  that  pE  shall  be 
less  than  B,  still  more  then  shall  p(mA  —  nB},  or 
pmA  —  pnB  be  less  than  B.  But  since  pFis  greater 
than  Q,  and  mP  —  nQ  is  by  hypothesis  greater  than 
Ft  still  more  shall  mpP  —  npQ  be  greater  than  Q. 
We  have  then,  if  our  last  supposition  be  correct,  some 
value  of  mp  and  np,  for  which 

mpA  —  npB  is  less  than  B, 
while 

mpP—npQ  is  greater  than  Q, 

or 

mpA  is  less  than  (np-\-  V)B, 

mpP  is  greater  than  (np  +  !)(?, 


PROPORTION.  26l 

which  is  contrary  to  our  first  hypothesis  respecting 
At  By  Py  said  Qy  that  hypothesis  being  Euclid's  defi- 
nition of  proportion,  from  which  if 

mpA  is  less  than  (np  -f-  V)B 

mpP  is  less  than  (np  +  \)Q. 

We  must  therefore  conclude  that  if  the  four  quantities 
Ay  By  Py  &&&  Q  satisfy  Euclid's  definition  of  propor- 
tion, then  m  and  n  may  be  so  taken  that  mA  —  nB  and 
mP  —  nQ  shall  be  as  small  as  we  please. 

Let 

mA  —  nB  =  E  and  £  =  nC 

mP  —  nQ  =  Fa.nd  F=nR. 
Then  mA  —  n(B  -f  C)  =  0 


and  since  E  and  F  can,  by  properly  assuming  m  and 
Hy  be  made  as  small  as  we  please,  much  more  can  the 
same  be  done  with  C  and  R,  consequently  we  can  pro- 
duce B  -f-  C  and  Q  -f-  R  as  near  as  we  please  to  B  and 
Qy  and  proportional  to  A  and  Pt  according  to  the 
common  arithmetical  notion.  In  the  same  way  it  may 
be  proved,  that  on  the  same  hypothesis  B  —  C  and 
Q  —  R  can  be  found  as  near  to  B  and  Q  as  we  please, 
and  so  that  A,  B  —  C,  P  and  Q  —  R  are  proportional 
according  to  the  ordinary  notion.  It  only  remains  to 
show  that  if  the  last-mentioned  property  be  assumed, 
Euclid's  definition  of  proportion  will  follow  from  it. 
That  is,  if  quantities  can  be  exhibited  as  near  to  P 
and  Q  as  we  please,  which  are  proportional  to  A  and 
By  according  to  the  ordinary  notion,  it  follows  that 


262  ON  THE  STUDY  OF  MATHEMATICS. 


For  let  B-\-  C  and  Q-\-  £  be  two  quantities,  such  that 


in  which,  by  the  hypothesis,  /and  g  can  be  so  taken 
that  C  and  ^  are  as  small  as  we  please.  We  have  al- 
ready shown  that  in  this  case  (m  and  n  being  any 
numbers  whatever)  mA  is  never  greater  or  less  than 
n(B  -f-  C),  without  mP  being  at  the  same  time  the 
same  with  regard  to  n(Q-\-  7?).  That  is,  if 

mA  is  greater  than  nB-\-nC, 
then 

mP  is  greater  than  nQ-\-  nR. 

Take  some  given*  values  for  m  and  n,  fulfilling  the 
first  condition  ;  then,  since  C  and  R  may  be  as  small 
as  we  please,  the  same  is  true  of  nC  and  nR;  if  then 

mA  is  greater  than  nB 

mP  is  greater  than  nQ. 

For  if  not,  let  mA=nJ3-{-x,  while  mP=nQ  —  y,  x 
and  y  being  some  definite  magnitudes.  Then  if 


which  last  equation  is  evidently  impossible  ;  therefore 
if  mA~>nB,  mP>nQ.     In  the  same  way  it  may  be 

*It  is  very  necessary  to  recollect  that  the  relations  just  expressed  are 
true  for  every  value  of  m  and  n  •  and  therefore  true  for  any  particular  case. 
In  this  investigation/"  and  g  may  both  be  very  great  in  order  that  C  and  R 
may  be  sufficiently  small,  and  we  must  suppose  them  to  vary  with  the  values 
we  give  to  C  and  /?,  or  rather  the  limits  which  we  assign  to  them  ;  but  m  and 


PROPORTION.  263 

proved  that  if  mA<^nB,  mP<nQ,  etc.,  so  that  Eu- 
clid's definition  is  shown  to  be  a  necessary  consequence 
of  the  one  proposed. 

The  definition  of  proportion  which  we  have  here 
given,  and  the  methods  by  which  we  have  established 
its  identity  with  the  one  in  use,  bear  a  close  analogy 
to  the  process  used  by  the  ancients,  and  denominated 
by  the  moderns  the  method  of  exhaustions.  We  have 
seen  that  the  common  definition  of  proportion  fails  in 
certain  cases  where  the  magnitudes  are  what  we  have 
called  incommensurable,  but  at  the  same  time  we 
have  shown  that  though  in  this  case  we  can  never 
take  m  and  n,  so  that  mA  =  nB,  or  mA  —  nB  =  0,  we 
can  nevertheless  find  m  and  n,  so  that  mA  shall  differ 
from  nB  by  a  quantity  less  than  any  which  we  please 
to  assign.  We  therefore  extend  the  definition  of  the 
word  proportion,  and  make  it  embrace  not  only  those 
magnitudes  which  fulfil  a  given  condition,  but  also 
others,  of  which  it  is  impossible  that  they  should  fulfil 
that  condition,  provided  always,  that  whatever  magni- 
tudes we  call  by  the  name  of  proportionals,  they  must 
be  such  as  to  admit  of  other  magnitudes  being  taken 
as  near  as  we  please  to  the  first,  which  are  propor- 
tional, according  to  the  common  arithmetical  notion. 
It  is  on  the  same  principle  that  in  algebra  we  admit 
the  existence  of  such  a  quantity  as  1/2,  and  use  it  in 
the  same  manner  as  a  definite  fraction,  although  there 
is  no  such  fraction  in  reality  as,  multiplied  by  itself, 
will  give  2  as  the  product.  But,  however  small  a 


264  ON  THE  STUDY  OF  MATHEMATICS. 

quantity  we  may  name,  we  can  assign  a  fraction  which, 
multiplied  by  itself,  shall  differ  less  from  2  than  that 
quantity. 

Having  established  the  properties  of  rectilinear 
figures,  as  far  as  their  proportions  are  concerned,  it  is 
necessary  to  ascertain  the  properties  of  curvilinear 
figures  in  this  respect.  And  here  occurs  a  difficulty 
of  the  same  kind  as  that  which  met  us  at  the  outset, 
for  no  rectilinear  figure,  how  small  soever  its  sides 
may  be,  or  how  great  soever  their  number,  can  be 
called  curvilinear.  Nevertheless,  it  may  be  shown 
that  in  every  curve  a  rectilinear  figure  may  be  in- 
scribed, whose  area  and  perimeter  shall  differ  from 
the  area  and  perimeter  of  the  curve  by  magnitudes 
less  than  any  assigned  magnitudes.  The  circle  is  the 
only  curve  whose  properties  are  considered  in  elemen- 
tary geometry,  and  the  proposition  in  question  is  dis- 
cussed in  all  standard  treatises  on  geometry.  Indeed, 
for  this  or  any  other  curve  the  proposition  is  almost 
self-evident.  This  being  granted,  the  properties  of 
curvilinear  figures  are  established  by  help  of  the  fol- 
lowing theorem. 

If  Ay  B,  C,  and  D  are  always  proportional,  and  of 
these,  if  C  and  D  may  be  made  as  near  as  we  please  to 
P  and  Q,  than  which  they  are  always  both  greater  or 
both  less,  then  A,  B,  P,  and  Q  are  proportional. 

Let  C=P+P',  and  £>=Q-\-Qf,  where  by  hy- 
pothesis P  and  Q'  may  be  made  as  small  as  we  please, 
and  A,  £,  P+P,  and  Q+  Q'  are  proportionals.  If 


PROPORTION.  265 


A,  B,  P,  and  Q  are  not  proportionals,  let  P  and 
be  proportional  to  A  and  B.  Then,  since  A  and  B 
are  proportional  to  P-\-P'  and  Q-\-  Q',  and  also  to  P 
and  <2  +  ^?,  therefore 

JP+P':Q+  Qr.P-.Q  +  R 

in  which  all  the  magnitudes  are  of  the  same  kind. 
Now  let  P'  and  Q  be  so  taken  that  Q'  is  less  than  ^?, 
which  may  be  done,  since  by  hypothesis  Q'  can  be  as 
small  as  we  please.  Hence  Q-{-  @  is  less  than^-j-^, 
and  therefore  P-\-P'  is  less  than  P,  which  is  absurd. 
In  the  same  way  it  may  be  proved  that  P  is  not  to 
O  —  R  in  the  proportion  of  A  to  B,  and  consequently 
P  is  to  Q  in  the  proportion  of  A  to  B.  This  theorem, 
with  those  which  prove  that  the  surfaces,  solidities, 
areas,  and  lengths,  of  curve  lines  and  surfaces,  may 
be  represented  as  nearly  as  we  please  by  the  surfaces, 
etc.,  of  rectilinear  figures  and  solids,  form  the  method 
of  exhaustions.*  In  this  method  are  the  first  germs  of 
that  theory  which,  under  the  name  of  Fluxions,  or  the 
Differential  Calculus,  contains  the  principles  of  all 
the  methods  of  investigation  now  employed,  whether 
in  pure  or  mixed  mathematics. 

*For  a  classical  example,  see  Prop.  II.  of  the  twelfth  book  of  Euclid 
(Simson's  edition).  Consult  also  Beman  and  Smith's  Plane  and  Solid  Geom- 
etry (Ginn  &  Co.,  Boston),  pp.  144-145,  and  igo.—Ed. 


CHAPTER  XVII. 

APPLICATION   OF  ALGEBRA  TO  THE   MEASUREMENT 

OF   LINES,    ANGLES,    PROPORTION    OF 

FIGURES,  AND  SURFACES. 

WE  have  already  defined  a  measure,  and  have  no- 
ticed several  instances  of  magnitudes  of  one 
kind  being  measured  by  those  of  another.  But  the 
most  useful  measure,  and  that  with  which  we  are  most 
familiar,  is  number.  We  express  one  line  by  the  num- 
ber of  times  which  another  line  is  repeated  in  it,  or  if 
the  second  is  not  exactly  contained  in  the  first,  by  the 
greatest  number  of  the  second  contained  in  the  first, 
together  with  the  fraction  of  the  second,  which  will 
complete  the  first.  Thus,  suppose  the  line  A  contains 
B  m  times,  with  a  remainder  which  can  be  formed  by 
dividing  B  into  q  parts,  and  taking  p  of  them.  Then 
B  is  to  A  in  the  proportion  of  1  to  m  -\-  — ,  or  as  q  to 
mq-\-p,  and  if  B  be  a  fixed  line,  which  is  used  for  the 
comparison  of  all  lines  whatsoever,  then  the  line  A  is 
m  -f  — ,  or  —  ,  if  it  be  understood  that  for  every 

*  *  'h 

unit  in  m,  B  is  to  be  taken,  and  also  that  for  —  the 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  267 

same  fraction  of  B  is  to  be  taken  that  —  is  of  unity. 
In  this  case  B  is  called  the  linear  unit. 

But  here  we  suppose  that  a  line  B  being  taken, 
the  ratio  of  any  other  line  A  to  B  can  be  expressed  by 
that  of  the  whole  numbers  mq-\-  p  to  q,  which  we  have 
shown  in  some  cases  to  be  impossible.  If  we  take 
one  of  these  cases,  mA  —  nB,  though  it  can  never  be 
made  equal  to  nothing,  can  be  made  as  small  as  we 
please,  by  properly  assuming  m  and  n.  Let  mA  —  nB 

=  £,  then  A  =  —  B -\ ,  and  since  —  can  be  made  as 

m  tn  m 

small  as  we  please,  A  can  be  represented  as  nearly  as 

we  please  by  a  fraction  — ,  where  B  is  the  linear  unit. 
m 

Hence,  in  practice  an  approximation  may  be  found  to 
the  value  of  A,  sufficient  for  any  purpose  whatever, 
in  the  following  manner,  which  will  be  easily  under- 
stood by  the  student  who  has  a  tolerable  facility  in 
performing  the  operations  of  algebra.  Let 

A  contain  B,  p  times  with  a  remainder  P, 
B  contain  P,  q  times  with  a  remainder  Q, 
P  contain  Q,  r  times  with  a  remainder  R, 
and  so  on.     If  the  two  magnitudes  are  commensur- 
able, this  operation  will  end  by  one  of  the  remainders 
becoming  nothing.     For,  let  A  and  B  have  a  common 
measure  E,  then  P  has  the  same  measure,  for  P  is 
A  — pB,  of  which  both  A  and  pB  contain  E  an  exact 
number  of  times.     Again,  because  B  and  P  contain 
the  common  measure  E,  Q  has  the  same  measure, 
and  so  on.   All  the  remainders  are  therefore  multiples 


268  ON  THE  STUDY  OF  MATHEMATICS. 

of  E,  and  if  E  be  the  linear  unit,  are  represented  by 
whole  numbers.  Now,  if  a  whole  number  be  contin- 
ually diminished  by  a  whole  number,  it  must,  if  the 
operation  can  be  continued  without  end,  eventually 
become  nothing.  If,  therefore,  the  remainder  never 
disappears,  it  is  a  sign  that  the  magnitudes  A  and  B 
are  incommensurable.  Nevertheless,  approximate 
whole  numbers  can  be  found  whose  ratio  is  as  near  as 
we  please  to  the  ratio  of  A  and  B. 

From  the  suppositions  above  mentioned  it  appears 

that 

(a) 

(V) 

(V) 

00 

00 
etc.,  etc. 

Substitute  in  (£)  the  value  of  P  derived  from  (0),  find 
Q  from  the  result,  and  substitute  the  values  of  P  and 
Q  in  (t)  ;  find  a  value  of  R  from  the  result,  and  sub- 
stitute the  values  of  Q  and  R  in  O/),  and  so  on,  which 
give  the  following  series  of  equations  : 


*  Throughout  these  investigations  the  capital  letters  represent  the  lines 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  269 

On  inspection  it  will  be  found  that  the  coefficients 
of  A  and  B  in  these  equations  may  be  formed  by  a 
very  simple  law.  In  each  a  letter  is  introduced  which 
was  not  in  the  preceding  one,  and  every  coefficient  is 
formed  from  the  two  preceding,  by  multiplying  the 
one  immediately  preceding  by  the  new  letter,  and  ad- 
ding to  the  product  the  one  which  comes  before  that. 
Thus  the  third  coefficient  of  B  is  pqr-\-p-\-r;  the 
new  letter  is  r,  and  the  two  preceding  coefficients  are 
pq+l  and  /,  zn&  pqr+p  +  r  =  (pq-\-  V)r-\-p.  The 
remainders  enter  also  with  signs  alternately  positive 
and  negative.  Let  x,  x',  and  x"  be  the  «th,  (n  -(-  1)*, 
and  (»-f  2)*  numbers  of  the  series  /,  q,  r,  etc.,  and 
X,  X'y  and  X"  the  corresponding  remainders.  Let 
the  corresponding  equations  be 

a  A=b  B  +  X 

a'A-^b'B—X' 


Here  n  must  be  supposed  odd,  since,  were  it  even, 
the  first  equation  would  be  aA  =  bB  —  X,  as  will  be 
seen  by  reference  to  the  equations  deduced.  Hence, 
from  the  law  of  formation  of  the  coefficients,  x"  being 
the  new  letter  in  the  last  equation, 

a"  =  a'x"  +  a 

b"  =  b'x"  -\-b. 

Eliminate  x"  from  these  two,  the  result  of  which 
is  a"  b'  —  a'b"  =  ab'  —  a'  b,  the  first  side  of  which  is 

themselves,  and  not  the  numbers  of  units,  which  represent  them,  while  the 
small  letters  are  whole  numbers,  as  in  the  last  chapter. 


270  ON  THE  STUDY  OF  MATHEMATICS. 

the  numerator  of  — , r.,  and  the  second  of  — , . 

a         a  a         a 

It  appears  then  that  — ,  is  either  greater  than  both 

b        ,  b"  ,a  ,       .         b'         b"       .  b'       b 

-  and  — ^  or  less  than  both,  since  — w  and  — 

a  a  a         a  a         a 

will  both  have  the  same  sign,  the  numerators  being 
the  same  and  the  denominators  positive.     It  may  also 

7ff  J  II 

be  proved  that  —  -,-  lies  between  —  and  —.  by  means  of 
a  a  a 

the  following  lemma. 

The  fraction must  lie  between  —  and  — ;  for 

P+9  P  q 

let  —  be  the  greater  of  the  two  last,  or  —  >  — ,  then 
/  P       9 

tno        Tip  on  <j  n 

or  -  L>  — — ,  or  —  >  — ,  and  1  4-— >  1  -\ — ; 

mp        mp         p       m  p  m 

1  -h- 
therefore  1  _!  ^  is  less  than  unity,   and  any  fraction 

multiplied  by  this  is  diminished.     But 
H  is  ^  X  T^-, 


and  is  therefore  less  than  — ,  the  greater  of  the  two. 

P 
In  the  same  way  it  may  be  proved  to  be  greater  than 

-,  the  least  of  the  two. 
? 

h"  hf  v"     I      A 

This  being  premised,  since  —r,  =  —. — r, ,  it  lies 

a          ax   -\-a 

b'x"        ,b.  b'        .  b 

between  — -. — Tl  and  —  or  between  — ;  and  — . 
a  x  a  a  a 

Call  the  coefficients  of  A  and  B  in  the  series  of 


equations,  a\t  a?,  etc.,  b\t  b^  etc.,  and  form  the  series 

of  fractions  — ,  — ,  — ,  etc.     The   two  first  of   these 
a\     a<2     a$ 

be  ^  and  ,  of   which   the   second   is   the 

a 


will 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  271 

greater,  since  it  is  /  H  --  .  Hence  by  what  has  been 
proved  —  is  less  than  —  and  greater  than—;  and 

#3  #2  #1 

every  fraction  is  greater  or  less  than  the  one  which 
comes  before  it,  according  as  the  number  of  its  equa- 
tion is  even  or  odd.  Again,  as  the  numerator  of  the 

difference  of  two  successive  fractions  -777  and  -77,  is  the 
r  b  b 

same  as  that  of  —,  and  T,  whatever  the  numerator  of 
o  b 

the  first  difference  is,  the  same  must  be  that  of  the 
second,  third,  etc.,  and  of  all  the  rest.  But  the  nu- 

merator of  the  difference  of  4~  and  is  1;  there- 

1  ?  b' 

fore  either  ab'  —  a'b,  or  a'b  —  ab',  is  1  according  as  —  -t 

7  a 

or  —  is  the  greater  of  the  two,  that  is  according  as  n 
is  odd  or  even.*  Now  since  the  «th  and  (n-\-  l)th  equa- 
tions, n  being  odd,  are 


and  a'A=b'B—X'; 
by  eliminating  A  we  have 

(a  V  —  a'b~)  B  =  a'X+  aX! 
or  £=a'X+aX' 

since  ab'  —  a'b  =  1  ;  and  since  the  remainders  decrease 
and   the   coefficients    increase,    a'^>  a    and    X*>  X', 


.     Tt 

whence  2  aX'  <  a'X  -f  aX\  or  2aX'<£  and  X'<-=-; 

&& 

the  remainder  therefore  which  comes  in  the  (n  -{-  1)* 
equation  is  less  than  the  part  of  B  arising  from  divid- 
ing it  into  twice  as  many  equal  parts  as  there  are 

*We  might  say  that  ab'  —  a'b  is  alternately  +  i  and  —  i;  but  we  wish  to 
avoid  the  use  of  the  isolated  negative  sign. 


272  ON  THE  STUDY  OF  MATHEMATICS. 

units  in  the  «th  coefficient  of  A  ;  and   as  this  number 
of  units  may  increase   to   any  amount  whatever,   by 

T> 

carrying  the  process  far  enough,  —  -  may  be  made  as 

Act 

small  as  we  please,  and  a  fortiori,  the  remainders  may 
be  made  as  small  as  we  please. 

The  same  theorem  may  be  proved  in  a  similar 
way,  if  we  begin  at  an  even  step  of  the  process.  Re- 
suming the  equations 

a  A=b  B  +  X 

a'  A=b'  B—X' 


we  obtain  from  the  second, 


T>  Vt  T> 

and  since  Jf  <  —  ,    —  7  <  ^  --  ;,  or  if  B  be  taken  as 
&a      a         A  a  a 

ji 

the  linear  unit,  —  will  express  the  line  A  with  an  error 

1       a 

less  than  -^  —  -,,  which  last  may  be  made  as  small  as 
£  a  a 

we  please  by  continuing  the  process. 

7  Jt 

It  is  also  evident  that  —  is  too  small,  while  —  is 
a  a 

too   great  ;    and    since  X  and  X'   are   less   than   B, 
aA<bB-\-B,oic  —    -  is  too  great,  while  a'A>b'B  —  B, 

or  -  —  ,    is   too  small.     Again,    A  --  B=—   and 
b'       a            X'  a  a 

-j  B  —  A=—r.  Now   X'  <X  and   tf'>0;    whence 
a                       a 

yf  y  jf 

—  -j  <  —  ;  that  is,  —  ,  B  exceeds  A  by  a  less  quantity 

than  —  B  falls  short  of  it,  so  that  —  r  is  a  nearer  repre- 

a  b 

sentation  of  A  than  —  ,  though  on  a  different  side  of  it. 
a 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  273 

We  have  thus  shown  how  to  find  the  representa- 
tion of  a  line  by  means  of  a  linear  unit,  which  is  in- 
commensurable with  it,  to  any  degree  of  nearness 
which  we  please.  This,  though  little  used  in  prac- 
tice, is  necessary  to  the  theory  ;  and  the  student  will 
see  that  the  method  here  followed  is  nearly  the  same 
as  that  of  continued  fractions  in  algebra.* 

We  now  come  to  the  measurement  of  an  angle  ; 
and  here  it  must  be  observed  that  there  are  two  dis- 
tinct measures  employed,  one  exclusively  in  theory, 
and  one  in  practice.  The  latter  is  the  well-known  di- 
vision of  the  right  angle  into  90  equal  parts,  each  of 
which  is  one  degree  ;  that  of  the  degree  into  60  equal 
parts,  each  of  which  is  one  minute  ;  and  of  the  minute 
into  60  parts,  each  of  which  is  one  second.  On  these 
it  is  unnecessary  to  enlarge,  as  this  division  is  perfectly 
arbitrary,  and  no  reason  can  be  assigned,  as  far  as  the- 
ory is  concerned,  for  conceiving  the  right  angle  to  be 
so  divided.  But  it  is  far  otherwise  with  the  measure 
which  we  come  to  consider,  to  which  we  shall  be  nat- 
urally led  by  the  theorems  relating  to  the  circle.  As- 
sume any  angle,  A  OB,  as  the  angular  unit,  and  any 
other  angle,  AOC(¥\g.  n).  Let  r  be  the  numberf  of 
linear  units  contained  in  the  radius  OA,  and  /  and  s 
the  lengths,  or  number  of  units  contained  in  the  arcs 
AB  and  AC.  Then  since  the  angles  A  OB  and  AOC 

*  See  Lagrange'  s  Elementary  Mathematics  (Chicago,  1898),  p.  2  et  seq.— Ed. 

t  It  must  be  recollected  that  the  word  number  means  both  whole  and 
fractional  number. 


274  ON  THE  STUDY  OF  MATHEMATICS. 

are  proportional  to  the  arcs  AB  and  AC,  or  to  the 
numbers  /  and  s,  we  have 

Angle  A  OC  is  -  of  the  angle  A  OB  ; 

and  the  angle  A  OB  being  the  angular  unit,  the  num- 
ber —  is  that  which  expresses  the  angle  A  OC.  This 
number  is  the  same  for  the  same  angle,  whatever 
circle  is  chosen  ;  in  the  circle  FD  the  proportion  of 


Fig.  ii. 

the  arcs  DE  and  DF  is  the  same  as  that  of  AB  and 
ACi  for  since  similar  arcs  of  different  circles  are  pro- 
portional to  their  radii, 

AB  :  DE  : :  OA  :  OD 
Also  AC:DF::OA:OD 
.-.  ABi>DE'.:AC:DF; 

therefore  the  proportion  of  DF  to  DE  is  that  of  s  to  /, 
and  -  is  the  measure  of  the  angle  DOF, — DOE  being 
the  unit,  as  before.  It  only  remains  to  choose  the 
angular  unit  A  OB,  and  here  that  angle  naturally  pre- 
sents itself,  whose  arc  is  equal  to  the  radius  in  length. 
This,  from  what  is  proved  in  Geometry,  will  be  the 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  275 

same  for  all  circles,  since  in  two  circles,  arcs  which 
have  the  same  ratio  (in  this  case  that  of  equality)  to 
their  radii,  subtend  the  same  angle.  Let  /  =  r,  then 
—  is  the  number  corresponding  to  the  angle  whose  arc 
is  s.  This  is  the  number  which  is  always  employed 
in  theory  as  the  measure  of  an  angle,  and  it  has  the 
advantage  of  being  independent  of  all  linear  units ; 
for  suppose  s  and  r  to  be  expressed,  for  example,  in 
feet,  then  12  s  and  12  rare  the  numbers  of  inches  in 

the  same  lines,  and  by  the  common  theory  of  frac- 

s        12  s 
tions  —  =  -£-  .     Generally,  the  alteration  of  the  unit 

does  not  affect  the  number  which  expresses  the  ratio 
of  two  magnitudes.  When  it  is  said  that  the  angle 
=  — -j-. — ,  it  is  only  meant  that,  on  one  particular  sup- 
position, (namely,  that  the  angle  1  is  that  angle  whose 
arc  is  equal  to  the  radius,)  the  number  of  these  units 
in  any  other  angle  is  found  by  dividing  the  number  of 
linear  units  in  its  arc  by  the  number  of  linear  units  in 
the  radius.  It  only  remains  to  give  a  formula  for  find- 
ing the  number  of  degrees,  minutes,  and  seconds  in 
an  angle,  whose  theoretical  measure  is  given.  It  is 
proved  in  geometry  that  the  ratio  of  the  circumference 
of  a  circle  to  its  diameter,  or  that  of  half  the  circum- 
ference to  its  radius,  though  it  cannot  be  expressed 
exactly,  is  between  3.14159265  and  3.14159266.  Tak- 
ing the  last  of  these,  which  will  be  more  than  a  suffi- 
cient approximation  for  our  purpose,  it  follows  that 
the  radius  being  r,  one-half  of  the  circumference  is 


276  ON  THE  STUDY  OF  MATHEMATICS. 

r  X  3.14159266  ;  and  one-fourth  of  the  circumference, 
or  the  arc  of  a  right  angle,  is  rX  1.57079633.  Hence 
the  number  of  units  above  described,  in  a  right  angle, 

is  — -p —  ,  or  1.57079633.    And  the  number  of  seconds 
radius 

in  a  right  angle  is  90  X  60  X  60,  or  324000.  Hence  if 
3-  be  an  angle  expressed  in  units  of  the  first  kind,  and 
A  the  number  of  seconds  in  the  same  angle,  the  pro- 
portion of  A  to  324000  will  also  be  that  of  5  to 
1.57079633.  To  understand  this,  recollect  that  the 
proportion  of  any  angle  to  the  right  angle  is  not  al- 
tered by  changing  the  units  in  which  both  are  ex- 
pressed, so  that  the  numbers  which  express  the  two 
for  one  unit,  are  proportional  to  the  like  numbers  for 
another. 

Hence  A  :  324000  : :  S  : 1.57079633  : 
324000 

Of   A  —  -^    e-rrnrro^oo   X       ' 

1. 57079633 
or  A  =206265  X  $,  very  nearly. 

Suppose,  for  example,  the  number  of  seconds  in  the 
theoretical  unit  itself  is  required.  Here  $  =  1  and 
^==206265;  similarly  if  A  be  1,  5=  ,  which 

is  the  expression  for  the  angle  of  one  second  referred 
to  the  other  unit.  In  this  way,  any  angle,  whose 
number  of  seconds  is  given,  may  be  expressed  in 
terms  of  the  angle  whose  arc  is  equal  to  the  radius, 
which,  for  distinction,  might  be  called  the  theoretical 
unit.*  This  unit  is  used  without  exception  in  analysis  ; 

•Also  called  a  radian.    See  Beman  and  Smith's  Geometry,  p. 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  277 

thus,  in  the  formula,  for  what  is  called  in  trigonom- 
etry the  sine  of  x,  viz.  : 


-,  etc. 


If  x  be  an  angle  of  one  second,  it  is  not  1  which  must 
be  substituted  for  x,  but  2Q6265' 

The  number  3.14159265,  etc.,  is  called  ;r,  and  is 
the  measure,  in  theoretical  units,  of  two  right  angles. 
Also  ^  is  the  measure  of  one  right  angle  ;  but  it  must 

2 

not  be  confounded,  as  is  frequently  done,  with  90°. 
It  is  true  that  they  stand  for  the  same  angle,  but  on 
different  suppositions  with  respect  to  the  unit  ;  the 
unit  of  the  first  being  very  nearly  ^  —  ^  times  that  of 
the  second. 

There  are  methods  of  ascertaining  the  value  of 
one  magnitude  by  means  of  another,  which,  though  it 
varies  with  the  first,  is  not  a  measure  of  it,  since  the 
increments  of  the  two  are  not  proportional  ;  for  exam- 
ple, when,  if  the  first  be  doubled,  the  second,  though 
it  changes  in  a  definite  manner,  is  not  doubled.  Such 
is  the  connexion  between  a  number  and  its  common 
logarithm,  which  latter  increases  much  more  slowly 
than  its  number  ;  since,  while  the  logarithm  changes 
from  0  to  1,  and  from  1  to  2,  the  number  changes 
from  1  to  10,  and  from  10  to  100,  and  so  on. 

Now,  of  all  triangles  which  have  the  same  angles, 
the  proportions  of  the  sides  are  the  same.  If,  there- 
fore, any  angle  CAB  be  given,  and  from  any  points 


278 


ON  THE  STUDY  OF  MATHEMATICS. 


B,  B ',  B" ,  etc.,  in  one  of  its  sides,  and  b,  b ',  etc.,  in 
the  other,  perpendiculars  be  let  fall  on  the  remaining 
side,  the  triangles  BAG,  B'AC',  bAc,  etc.,  having  a 
right  angle  in  all,  and  the  angle  A  common,  are  equi- 
angular ;  that  is,  one  angle  being  given,  which  is  not 
a  right  angle,  the  proportions  of  every  right-angled 
triangle  in  which  that  angle  occurs  are  given  also  ; 
and,  vice  versa,  if  the  proportion,  or  ratio  of  any  two 
sides  of  a  right-angled  triangle  are  given,  the  angles 
of  the  triangle  are  given. 

B" 


Vc" 


To  these  ratios  names  are  given  ;  and  as  the  ra- 
tios themselves  are  connected  with  the  angles,  so  that 
one  of  either  set  being  given,  viz.,  ratios  or  angles, 
all  of  both  are  known,  their  names  bear  in  them  the 
name  of  the  angle  to  which  they  are  supposed  to  be 

L  BC        side  opposite  to  A    . 

referred.    Thus,  — =,  or  — -. — ^-. — ,  is  called 

AB*  hypothenuse 

...     AC         side  opposite  to  B 

the  sine  of  A  :  while  -^=- ,  or  — : — ^ ,  or  the 

AB  hypothenuse 

sine  of  B,  the  complement*  of  A,  is  called  the  cosine 

*When  two  angles  are  together  equal  to  a  right  angle,  each  is  called  the 
complement  of  the  other.  Generally,  complement  is  the  name  given  to  one 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC. 


279 


of  A.    The  following  table  expresses  the  names  which 

,       .         .       BC  AC  BC  AC  AB 
ra  given  to  the  six  ratios, -jg  —  —  —  —  and 

-=^,  relatively  to  both  angles,  with  the  abbreviations 
-£>G 

made  use  of.  The  terms  opp.,  adj.,  and  hyp.,  stand 
for,  opposite  side,  adjacent  side,  and  hypothenuse,  and 
refer  to  the  angle  last  mentioned  in  the  table. 


THE 
RATIO 

IS  THE 

BEING 

OR 

BEING 

THESE  ARE  WRITTEN 

BC 

AB 

sine  of  A 

OPP- 

hyp. 

cosine  of  B 

adj. 
hyp. 

sin  A 

cos  B 

AC 
~AB 

cosine  of  A 

adj. 
hyp. 

sine  of  B 

OPP- 

hyp. 

cos  A 

sin  B 

BC 
AC 

tangent  of  A 

opp. 
adj. 

cotangent  of  B 

adj. 
opp. 

tan  A 

cot  B 

AC 
^C 

cotangent  of  A 

adj. 
opp. 

tangent  of  B 

opp. 
adj. 

cot  A 

tan  B 

AB 

~AC 

secant  of  A 

hyp. 
adj. 

cosecant  of  B 

hyp. 
opp. 

sec  A 

cosec.# 

AB 
BC 

cosecant  of  A 

hyp- 

opp. 

secant  of  B 

hyp. 
adj. 

cosecA 

sec£ 

If  all  angles  be  taken,  beginning  from  one  minute, 
and  proceeding  through  2',  3',  etc.,  up  to  45°,  or  2700', 
and  tables  be  formed  by  a  calculation,  the  nature  of 
which  we  cannot  explain  here,  of  their  sines,  cosines, 
and  tangents,  or  of  the  logarithms  of  these,  the  pro- 
portions of  every  right-angled  triangle,  one  of  whose 
angles  is  an  exact  number  of  minutes,  are  registered. 


part  of  a  whole  relatively  to  the  rest.    Thus,  10  being  made  of  7  and  3,  7  is 
the  complement  of  3  to  10. 


280  ON  THE  STUDY  OF  MATHEMATICS. 

We  say  sines,  cosines,  and  tangents  only,  because  it 
is  evident,  'from  the  table  above  made,  that  the  co- 
secant, secant,  and  cotangent  of  any  angle,  are  the 
reciprocals  of  its  sine,  cosine,  and  tangent,  respec- 
tively. Again,  the  table  need  only  include  45°,  in- 
stead of  the  whole  right  angle,  because,  the  sine  of  an 
angle  above  45°  being  the  cosine  of  its  complement, 
which  is  less  than  45°,  is  already  registered.  Now,  as 
all  rectilinear  figures  can  be  divided  into  triangles, 
and  every  triangle  is  either  right-angled,  or  the  sum 
or  difference  of  two  right-angled  triangles,  a  table  of 
this  sort  is  ultimately  a  register  of  the  proportions  of 
all  figures  whatsoever.  The  rules  for  applying  these 
tables  form  the  subject  of  trigonometry,  which  is  one 
of  the  great  branches  of  the  application  of  algebra  to 
geometry.  In  a  right-angled  triangle,  whose  angles 
do  not  contain  an  exact  number  of  minutes,  the  pro- 
portions may  be  found  from  the  tables  by  the  method 
explained  in  Chapter  XI.  of  this  treatise.  It  must  be 
observed,  that  the  sine,  cosine,  etc.,  are  not  measures 
of  their  angle  ;  for,  though  the  angle  is  given  when 
either  of  them  is  given,  yet,  if  the  angle  be  increased 
in  any  proportion,  the  sine  is  not  increased  in  the 
same  proportion.  Thus,  sin  2A  is  not  double  of  sin  A. 
The  measurement  of  surfaces  may  be  reduced  to 
the  measurement  of  rectangles ;  since  every  figure 
may  be  divided  into  triangles,  and  every  triangle  is 
half  of  a  rectangle  on  the  same  base  and  altitude.  The 
superficial  unit  or  quantity  of  space,  in  terms  of  which 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  28 1 

it  is  chosen  to  express  all  other  spaces,  is  perfectly 
arbitrary  ;  nevertheless,  a  common  theorem  points  out 
the  convenience  of  choosing,  as  the  superficial  unit, 
the  square  on  that  line  which  is  chosen  as  the  linear 
unit.  If  the  sides  of  a  rectangle  contain  a  and  b  units, 
the  rectangle  itself  contains  ab  of  the  squares  de- 
scribed on  the  unit.  This  proposition  is  true,  even 
when  a  and  b  are  fractional.  Let  the  number  of  units 

in  the  sides  be  —  and  — ,  and  take  another  unit  which 
1  n  q 

is  —  of  the  first,  or  is  obtained  by  dividing  the  first 
nq 

unit  into  nq  parts,  and  taking  one  of  them.  Then, 
by  the  proposition  just  quoted,  the  square  described 
on  the  larger  unit  contains  nqy^nq  of  that  described 
on  the  smaller.  Again,  since  —  and  —  are  the  same 

fractions  as  —  and  — ,  they  are  formed  by  dividing 
nq  nq 

the  first  unit  into  nq  parts,  and  taking  one  of  these 
parts  mq  and  np  times  ;  that  is,  they  contain  mq  and 
np  of  the  smaller  unit ;  and,  therefore,  the  rectangle 
contained  by  them,  contains  mqY^np  of  the  square 
described  on  the  smaller  unit.  But  of  these  there  are 

nqY^nq  in  the  square  on  the  longer  unit ;  and,  there- 

,         mqy^np         mp  Y.nq         mp     . 
fore,  -       —  — ,  or  — —     - ,  or  — ,   is  the  number  of 
nqy^nq  nqy^nq          nq 

the  larger  squares  contained  in  the  rectangle.  But 
-£-  is  the  algebraical  product  of  —  and  — .  This  prop- 
osition is  true  in  the  following  sense,  where  the  sides 
of  the  rectangle  are  incommensurable  with  the  unit. 
Whatever  the  unit  may  be,  we  have  shown  that,  for 


282 


ON  THE  STUDY  OF  MATHEMATICS. 


any  incommensurable  magnitude,  we  can  go  on  finding 

b  and  a,  two  whole  numbers,  so  that  —  is  too  little,  and 

b+ 1  a 

-  too  great  :  until  a  is  as  great  as  we  please.    Let 

AB  and  A  C  be  the  sides  of  a  rectangle  AK,  and  let 
them  be  incommensurable  with  the  unit  M.     Let  the 

lines  AF  and  AG,  containing  —  and  -      -  units,  be 

a  a 

respectively  less  and  greater  than  AC;  and  let  AD 
and  AE,  containing  —  and  — —  units,  be  respectively 


K 


H 


Fig.  13. 


F    C     G 


less  and  greater  than  AB ;  and  complete  the  figure. 
The   rectangles   AH  and   AI  contain,    respectively, 

—  X  —T,   and X  — j —   square   units,*  and   the 

ad  a  d 

first  is  less  than  the  given  rectangle,  and  the  second 
greater ;  consequently  the  given   rectangle  does  not 
differ  from  either,  so  much  as  they  differ  from  one 
another.     But  the  difference  of  AH  and  AI  is 
!         be 


ad 


—  — -,,  or 


ad' 


ad 


*"  Square  unit"  is  the  abbreviation  of  "square  described  on  the  unit." 


ON  THE  APPLICATION  OF  ALGEBRA,   ETC.  283 

6r  1 A  +  1 .1  4.  -L 

d  a         ad'    ad' 

Proceed  through  two,*  four,  six,  etc.,  steps  of  the 
approximation.  The  linear  unit  being  M,  the  results 

will  be  such,  that—  M will  be  always  less  than  A C, 
a  -1    z 

but  continually  approaching  to  it.     Hence  -=  —  M  is 

AC  a 

always  less  than  — =- ;  and  since  A  C  remains  the  same, 
a 

and  d  is  a  number  which  may  increase  as  much  as  we 

AC 

please,   by  carrying  on   the   approximation,  -  r  and 
I/  a 

a  fortiori  —  —  M  may  be  made  as  small  a  line  as  we 

1    b 

please ;  that  is,  —  —  may  be   made   as   small   as   we 
da     j    £ 

please,   and   so   may T  in  the  same  manner.     Also 

•i  ad 

—j  may  be  made  as  small  as  we  please ;  and  there- 
ad  -i  i  -i  -i 

fore,  also,  the  sum  — 1 r  -I -j.   But  this  num- 

d  a         ad        ad 

ber,  when  the  unit  is  the  square  unit,  represents  the 
difference  of  the  rectangles  AH  and  AI,  and  is  greater 
than  the  difference  of  AK  and  AI ;  therefore,  the  ap- 
proximate fractions  which  represent  AC  and  AB  may 
be  brought  so  near,  that  their  product  shall,  as  nearly 
as  we  please,  represent  the  number  of  square  units  in 
their  rectangle. 

In  precisely  the  same  manner  it  may  be  proved, 
that  if  the  unit  of  content  or  solidity  be  the  cube  de- 
scribed on  the  unit  of  length,  the  number  of  cubical 
units  in  any  rectangular  parallelepiped,  is  the  product 

*This  is  done,  because,  by  proceeding  one  step  at  a  time,  —  is  alternately 
too  little  and  too  great  to  represent  AC;  whereas  we  wish  the  successive 
steps  to  give  results  always  less  than  AC. 


284  ON  THE  STUDY  OF  MATHEMATICS. 

of  the  number  of  linear  units  in  its  three  sides,  whether 
these  numbers  be  whole  or  fractional ;  and  in  the  sense 
just  established,  even  if  they  be  incommensurable  with 
the  unit. 

These  algebraical  relations  between  the  sides  and 
content  of  a  rectangle  or  parallelepiped  were  observed 
by  the  Greek  geometers  ;  but  as  they  had  no  distinct 
science  of  algebra,  and  a  very  imperfect  system  of 
arithmetic,  while,  with  them,  geometry  was  in  an  ad- 
vanced state ;  instead  of  applying  algebra  to  geom- 
etry, what  they  knew  of  the  first  was  by  deduction 
from  the  last :  hence  the  names  which,  to  this  day, 
are  given  to  aa,  aaa,  abt  which  are  called  the  square 
of  a,  the  cube  of  a,  the  rectangle  of  a  and  b.  The  stu- 
dent is  thus  led  to  imagine  that  he  has  proved  that 
square  described  on  the  line  whose  number  of  units 
is  a,  to  contain  aa  square  units,  because  he  calls  the 
latter  the  square  of  a.  He  must,  however,  recollect, 
that  squares  in  algebra  and  geometry  mean  distinct 
things.  It  would  be  much  better  if  he  would  accus- 
tom himself  to  call  a  a  and  aaa  the  second  and  third 
powers  of  a,  by  which  means  the  confusion  would  be 
avoided.  It  is,  nevertheless,  too  much  to  expect  that 
a  method  of  speaking,  so  commonly  received,  should 
ever  be  changed  ;  all  that  can  be  done  is,  to  point  out 
the  real  connexion  of  the  geometrical  and  algebraical 
signification.  This,  if  once  thoroughly  understood, 
will  prevent  any  future  misconception. 


INDEX. 


Addition,  23,  67. 

Algebra,  notation  of,  55  et  seq.;  ele- 
mentary rules  of,  67  et  seq.;  advice 
on  the  study  of,  53,  54,  62,  175  et 
seq.;  nature  of  the  reasoning  in, 
192 ;  applied  to  the  measurement 
of  lines,  angles,  proportion  of  fig- 
ures and  surfaces,  266-284. 

Algebraically  greater,  144-145. 

Algebras,  bibliographical  list  of,  188- 
189. 

Analogy,  in  language  of  algebra,  79. 

Angle,  definition  of,  196  et  seq.,  238; 
measure  of,  273  et  seq, 

Angular  units,  275-276. 

Approximations,  48  et  seq.,  130;  171 
et  seq.,  242  et  seq.;  267;  281  et  seq. 

Arrangment  of  algebraical  expres- 
sions, 73. 

Arithmetic,  elementary  rules  of,  20 
et  seq.;  compared  with  algebra,  76. 

Arithmetical,  notation,  n  et  seq.;  no- 
tion of  proportion,  244  et  seq. 

Assertions,  logical,  203  et  seq. 

Assumptions,  231  232. 

Axioms,  208,  231  et  seq. 

Babbage,  168,  174. 

Bagay,  168. 

Bain,  212. 

Baltzer,  R.,  189. 

Beman,  W.  W.,  188,  265,  276. 

Bertrand,  239. 

Biermann,  O.,  189. 

Binomial  theorem,  exercises  in,  177  et 

seq. 

Bolyai,  232. 
Bosanquet,  212. 


Bourdon,  188. 

Bourgeois  gentilhomme,  the,  230. 

Bourlet,  C.,  188. 

Brackets,  21. 

Bradley,  212. 

Bremiker,  168. 

Bruhns,  168. 

Burnside,  W.  S.,  189. 

Caesar,  2,  229. 

Caillet,  168. 

Callet,  168. 

Carus,  Paul,  232. 

Change  of  algebraical  form,  105  et 

seq. 

Chrystal,  Prof.,  189. 
Cipher,  16. 

Circulating  decimals,  51. 
Clifford,  232. 
Coefficient,  60. 
Collin,  J.,  188. 
Commercial  arithmetic,  53. 
Comparison  of  quantities,  244  et  seq, 
Computation,  180. 
Comte,  187. 
Condillac,  187. 
Continued  fractions,  267-273. 
Contradictory,  205. 
Contraries,  205. 

Convergent  fractions,  269  et  seq. 
Converse,  205. 
Copula,  203. 
Counting,  13  et  seq. 
Courier,  problem  of  the  two,  112  et 

seq. 
Cube,  the  term,  284. 

D'Alembert,  187. 
Dauge,  F.,  187. 


286 


ON  THE  STUDY  OF  MATHEMATICS. 


Decimal,  system  of  numeration,  14  et 

seq. ;  point,  43 ;  fractions,  42-54. 
Definition,  u,  207. 
Delbceuf,  J.,  232. 
Demonstration,    mathematical,  4   et 

seq.,  184;  inductive,  179. 
De  Morgan,  187. 
Descartes,  37. 
Differential  calculus,  265. 
Diminution,  not  necessarily  without 

limit,  251. 
Diophantus,  186. 
Direction,  196. 
Direct  reasoning,  226. 
Discovery,  progress  of,  dependent  on 

language,  37. 

Division,  23  et  seq.,  38,  75,  165-167. 
Duhamel,  72,  187,  232. 
Duhring,  187. 
Duodecimal  system,  19. 

E;:^iisJi  Cyclopedia,  174,  187. 

Equations,  of  the  first  degree,  90-102 ; 
of  the  second  degree,  129  et  seq. ; 
identical,  90;  of  condition,  91,  96 
et  seq.;  reducing  problems  to,  92 
et  seq. 

Errors,  in  mathematical  computa- 
tions, 48  et  seq. ;  in  algebraical 
suppositions, corrected  by  a  change 
of  signs,  106  et  seq. 

Euclid,  4,  37,  181,  234-235,  265;  his 
theory  of  proportion,  240, 258  et  seq. 

Exhaustions,  method  of,  263-265. 

Exponents.     See  Indices. 

Experience,  mathematical,  104. 

Expressions,  algebraical,  59. 

Extension  of  rules  and  meanings  of 
terms,  33  et  seq.,  80-82, 143-145,  163. 

Factoring,  132  et  seq.,  160. 

Figures,  logical,  216  ei  seq. 

Fine,  H.  B.,  189. 

Fisher  and  Schwatt,  189. 

Fluxions,  265. 

Form,  change  of  in  algebraical  ex- 
pressions, 105  et  seq.,  117  et  seq. 

Formula?,  important,  88-89,  99.  141- 
142,  163-167. 

Fowler,  T.,  212. 

Fractions,  arithmetical,  30  et  seq., 


75;    decimal,   42;    continued,   267- 

273;  singular  values  of,  123  et  seq.; 

evanescent,    126-128;     algebraical, 

75,  87-89,  97-99. 

Fractional  exponents,  163  et  seq.,  185. 
French  language,  188. 
Frend,  71,  foot-note. 
Freycinet,  187. 

Geometry,  study  of,  4  et  seq.;  defini- 
tions and  study  of,  191  et  seq.;  ele- 
mentary ideas  of,  193  et  seq. 

Geometrical  reasoning  and  proof,  203 
et  seq.,  220  et  seq. 

German  language,  188. 

Grassmann,  232. 

Greatest  common  measure,  25  et  seq., 
86,  267  et  seq. 

Greater  and  less,  the  meaning  of,  144, 

Greatness  and  smallness,  170. 

Halsted,  232. 
Harlot,  38. 
Haskell,  168. 
Hassler,  168. 
Helmholtz,  232. 
Hindu  algebra,  186. 
Hirsch,  188. 
Holzmiiller,  G.,  189. 
Hutton,  168. 
Hyde,  232. 
Hypothesis,  208. 

Identical  equations,  90. 

Imaginary  quantities,  151  et  seq. 

Impossible  quantities,  149  et  seq. 

Incommensurables,  246  et  seq.,  281  et 
seq. 

Increment,  169. 

Indirect  reasoning,  226. 

Indeterminate  problems,  101. 

Indices,  theory  of,  60, 158  et  seq.,  166, 
185. 

Induction,  mathematical,  104, 179,183. 

Inductive  reasoning,  219. 

Infinite  quantity,  meaning  of,  123  et 
seq. 

Infinite  spaces,  compared,  235  et  seq. 

Instruction,  principles  of  natural,  21 
et  seq.;  faulty,  182;  books  on  math- 
ematical, 187. 

Interpolation,  169-174. 


INDEX. 


287 


James,  W.,  232. 
Jevons,  212. 
Jodl,  F.,  232. 
Jones,  168. 

Keynes,  212. 

Lacroix,  187. 

Lagrange,  187. 

Laisant,  187. 

Lallande,  168. 

Language,  13,  37,  79. 

Laplace,  185. 

Laurent,  H.,  188. 

Least  common  multiple,  28. 

Leibnitz,  37. 

Line,  193,  242. 

Linear  unit,  267. 

Literal  notation,  57  et  seq. 

Lobachevski,  232. 

Locke,  9. 

Logarithms,  167  et  seq. 

Logic  of  mathematics,  203-230. 

Logics,  bibliographical  list  of,  212. 

Mach,  E.,  232. 

Mathematics,  nature,  object,  and 
utility  of  the  study  of,  i  et  seq.; 
language  of,  37  et  seq.;  advice  on 
study  of,  175  ;  philosophy  of,  187. 

Matthiessen,  189. 

Measures,  198,  266  et  seq. 

Measurement,  of  lines,  angles,  pro- 
portion of  figures,  and  suriaces, 
266-284. 

Measuring,  241  et  seq. 

Mill,  J.  S.,  212. 

Minus  quantities,  72. 

Mistaken  suppositions,  106  et  seq. 

Moods,  logical,  212  et  seq. 

Multiplication,  23  et  seq.,  34  et  seq., 
68  et  seq.,  164. 

Mysticism  in  numbers,  14. 

Negative,  quantities,  72;  sign,  iso- 
lated, 103  et  seq.,  181 ;  squares,  149, 
151;  indices,  166,  185. 

Netto,  E.,  189. 

Newton,  37,  185. 

Notation,  arithmetical,  decimal,  n 
et  seq.;  general  principle  of,  15  et 


seq.;  algebraical,  55  et  seq.;  79, 159, 
extension  of,  33,  80,  143,  163. 

Numbers,  representation  of,  15  et 
seq. 

Numeration,  systems  of,  14  et  seq. 

Numerically  greater,  144. 

Oliver,  Waite,  and  Jones,  189. 

Panton,  A.  W.,  189. 

Parallels,  theory  of,  181,  231-237. 

Particular  affirmative  and  negative, 
203. 

Perfect  square,  138. 

Petersen,  189. 

T,  277. 

Plane  surface,  195. 

Plato,  239. 

Poincare,  H.,  232. 

Point,  geometrical,  194-195. 

Postulate,  210. 

Powers,  theory  of,  158  et  seq. 

Predicate,  203. 

Premisses,  211  et  seq. 

Prime    numbers   and  factors,  254 

Problems,  reducing  of,  to  equations, 
92  et  seq.;  general  disciplinary  util- 
ity of,  95 ;  of  loss  and  gain  as  illus- 
trating changes  of  sign,  119 ;  of  the 
two  couriers,  112  et  seq. 

Proportions,  170;  theory  of,  240-265. 

Proportional  parts,  173. 

Propositions,  203  et  seq. 

Pythagorean  proposition,  221  et  seq, 

Quadratic,  equations,  129  et  seq.; 
roots,  discussion  of  the  character 
of,  137  et  seq. 

Radian,  276. 

Read,  Carveth,  212. 

Reasoning,  geometrical,  203  et  seq.; 

direct  and  indirect,  226. 
Reckoning,  13  et  seq. 
Riemann,  232. 
Roots,  129  et  seq.,  137  et  seq.,  158  et 

seq. 

Rules,  42;  mechanical,  184. 
Rules,  extension  of  meaning  of,  33, 

80,  143,  163. 
Russell,  B.  A.  W.,  232. 


288 


ON  THE  STUDY  OF  MATHEMATICS. 


Schlomilch,  O.,  189. 

Schron,  168. 

Schubert,  H.,  189. 

Schiiller,  W.  J.,  189. 

Self-evidence,  209. 

Serret,  J.  A.,  188. 

Sexagesimal  system  of  angular  meas- 
urement, 273. 

Shorthand  symbols,  55. 

Sidgwick,  212. 

Signs,  arithmetical  and  algebraical, 
20  et  seq. ;  55 ;  rule  of,  96,  186. 

Sigwart,  212. 

Simple  expression,  76,  77. 

Singular  values,  122  et  seq. 

Smith,  D.  E.,  IV.,  187,  265,  276. 

Solutions,  general  algebraical,  in. 

Square,  the  term,  282,  284. 

Stackel,  Paul,  232. 

Stallo,  J.  B.,  232. 

Straight  line,  12,  193. 

Subject,  203. 

Subtraction,  23. 

Subtractions,  impossible,  103-104. 

Surfaces,  measurement  of  incom- 
mensurable, 280  et  seq. 

Syllogisms,  210  et  seq. 

Symbols,  invention  of,  80-81.  See 
Signs. 

Syllabi,  mathematical,  186. 


Tables, mathematical, recommended, 
168. 

Tannery,  P.,  232. 

Taylor,  168. 

Terms,  geometrical  and  algebraical 
compared,  284. 

Theory  of  equations,  132  et  seq.,  179, 
190. 

Todhunter,  189. 

Triangles,  measurement  of  propor- 
tions of,  277  et  seq. 

Trigonometrical  ratios,  278  et  seq. 

Ueberweg,  212. 

Universal  affirmative  and  negative, 
203. 

Vega,  168. 
Venn,  212. 

Weber,  H.,  189. 
Wells,  168. 
Whately,  212. 
Whole  number,  76. 

Zero,  as  a  figure,  16;  its  varying  sig- 
nificance as  an  algebraical  icsult, 
122  et  seq.;  exponents,  81,  166. 


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029S5   4 


o 


T.APR  16 1971 


QA  De  Morgan,  Augustus 

11       On  the  study  and  difficulties 

D^6  of  mathematics  2d  reprint  ed. 

1902 


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